Jessica Fridrich

Rubik's cube 11/25/11 10:51 AM

My speed cubing page

This is a copy of Ernö Rubik's signature as it appears in my notebook. He signed it at the World Championship in Budapest in 1982

This system for advanced cubers and is not appropriate for a beginner. It is intended for those of you who can already solve the cube in a few minutes and want to get really fast. If you are a complete beginner, please, visit Jasmine's Beginner Solution.

My system for solving Rubik's cube Unique features The first two layers (additional useful hints and examples of how I solve the first two layers) The last layer 20 years of speed cubing (a short historical narrative) Watch me solving the cube Hints for speed cubing Customizing algorithms Multiple algorithms Finger shortcuts Move algorithms to your subconsciousness No delays between algorithms Faster twisting does not have to mean shorter times Preparing the cube for record times Hard work What are the limits of speed cubing?

Collections of various algorithms (by Mirek Goljan, mgoljan (AT) binghamton. edu) Swapping two edges and two corners

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Swaping two and two edges Twisting and moving corners and edges in one layer (by Mirek Goljan, mgoljan (AT) binghamton. edu) Pretty patterns by Mirek Goljan, mgoljan (AT) binghamton. edu

Richard Carr is THE expert on solving large cubes with a list of his record times. Richard can solve the cube blindfolded and willingly shares with us his method. I met Richard in April 2003 and he showed me his incredible skills in person.

Guus Razoux Schultz on speed cubing

The World Championship, Budapest 1982

Hana Bizek's cube art

Dutch Cube Day, October 6, 2002

San Francisco Cubing, January 19, 2003

The 2nd World Championship in Rubik's Cube in Toronto, August 23-24, 2003

Press&Sun Bulletin, Binghamton Sep 11, 2003

Cube links

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My system for solving Rubik's cube

Winter 1996/97: The system described here enabled me to win the First Czechoslovak Championship in Rubik's Cube, which took place in April 1982. When I was at my best, I routinely solved the cube in an average time of 17 seconds. At that time, I was actively using more than 100 algorithms, but the basic required minimum is 53 algorithms. Before I go on and describe the details of my system, I would like to express my thanks to Mike Pugh who retyped the algorithms from my old notebook to HTML and added nice graphics. His enthusiasm helped me to find the cube no less interesting than some 15+ years ago when I met it for the first time. Special thanks belong to Mirek Goljan, my 1982 finale rival, who kindly provided his enormous collection of algorithms as it appears here today.

There are a number of diferent systems suitable for speed cubing, but all can be roughly divided into two main categories: corners-edges and by-layers. My system belongs to the second category even though the first two layers are really formed at the same time rather than in sequence. The basic set of algorithms consists of 53 algorithms for the last layer and a couple of simple moves for the second layer together with a lot of experience. Most of the algorithms were developed by myself during the time period between the summer 1981 and the spring of 1983. However, other speed cubists, most noticeably Mirek Goljan, have also significantly contributed with important moves. Here is my system in a nut shell:

Average number Action description Time Result of moves

Place the four edges from the first layer 7 2 sec.

Place four blocks each consisting of one corner from the first layer and a 4 x 2 4 x 7 corresponding sec. edge from the second layer.

Simultaneously orient the corners AND

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edges 9 3 sec. so that the last layer has the required color (one algorithm out of 40).

Simultaneously permute the 8 cubes in the last layer without rotating corners or flipping 12 4 sec. edges (one algorithm out of 13).

TOTAL 56 17

Unique features

One of the unique features of this system is that the last layer is always solved using two algorithms of an average length of 9 and 12, which is very efficient. The average lengths are based on frequencies with which various orientations and permutations occur and on the length of algoritms for each position. Another interesting feature is that for the first two layers no lengthy algorithms are needed and you can use your intuition and utilize the specifics of the particular initial state and subsequent states of the cube.

The first two layers

In an attempt to make this description complete, I supplied several algorithms which can help you solve the first two layers. Although most of the algorithms will be obvious to an experienced speed cubist, some of them are less trivial and are in my opinion very valuable. In addition to that, one should always try to use the specifics of any given state of the cube rather than blindly apply the algorithms. For example, when two or three corners are already correctly placed, it may be advantageous to keep the last corner free and insert all middle cubes using the free corner. Actually, some speed cubists use this approach as their default. Alternatively, when accidentaly (or intentionally) two or more middle cubes happen to be positioned correctly, one can place the corners from the first layer via the free middle edge(s). All these moves and

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a lot of practice should enable you to solve the first two layers in about 10-12 sec. Of course, this requires a lot of practicing, but let us say that 15-20 sec. will be realistic for most folks.

Because I was receiving a lot of requests for "additional hints" and advice for the middle layer, I decided to include another section with practical advice for solving the middle layer. Here are a few examples of how I think out loud when doing the middle layer. I hope this will help you to master the system faster!

The last layer

Some systems for solving Rubik's cube "by-layers" divide the solution of the last layer into four stages: orient edges, place edges, orient corners, place corners. It is possible to group together two and two stages to speed up the process. It seems natural to orient and place edges in one move and then orient and place corners in the second one. However, this approach has one big disadvantage - it is very difficult to recognize various positions quickly. A better approach is to orient edges and corners at the same time and place all of them simultaneously. Convince yourself that there are 41 different orientations of the cubies in the last layer, and 14 different permutations of those 8 cubies. Here, we do not count symmetric positions or inverse (backwards) positions as different because they can be solved using one algorithm. Different orientations are easily recognizable by patterns formed by the color of the last layer and a brief look at the sides of the cube. There are two patterns "C", four "I", two "T", etc. Most of the permutations are also easily recognizable. Given an average twisting speed of three moves per second, one can solve the last layer in 3 + 4 seconds (based on the average number of moves).

In theory, we could come up with a much larger system of algorithms which would enable us to solve the last layer in one algorithm. However, the number of algorithms one would need to learn is 1211.

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Dear visitor, I am sure my narrative will occassionally bring a smile of disbelief to your faces today, but I want to truthfully describe the atmosphere of excitement and mystery as we, the old-timers, lived through the "good ol' days" of the early eighties when Rubik's cube was making headlines around the world.

The first time I met the cube "face-to-face" was when I was 16 years old in March 1981. I was hooked since the first moment I saw this absolutely unique combination of simplicity and ingenuity. There was no need to explain what needs to be done with it - a self-explanatory, remarkably difficult puzzle with a devilishly mysterious mechanism inside - a fascinating silent challenge. The owner of the cube was a 14-year old boy who could solve the cube in about a minute. He lent it to me for a few minutes just enough to assemble one face.

Although in March 1981 the cube was being sold by thousands in other countries and despite the fact that the invention took place in the neighboring country, it was impossible to buy the cube in Czech Republic. A classical example of how inefficient and impotent the Eastern Block economies were. I got my hands on a primitive solving system from a Russian magazine Kvant long before I actually owned the cube. I would analyze simple moves and their action on a piece of paper, trying to figure out algorithms based on the commutator principle. Then later in the spring, our local astronomy club leader bought the cube during his trip to Hungary. He was unable to solve it and could not find anybody who would put the cube back into its original state. With the help of the commutator principle and those "Russian" moves, I solved the cube for the first time. It took me several hours.

I desperately wanted to get my own cube but whoever was lucky enough to own it, would never sell it. So, I had to wait a little longer and finally got my first cube in July 1981. A French family was visiting my sister and their two teenage boys brought the cube with them. When they saw how attached I quickly became to the piece of plastic, they did not have the heart to take http://ws2.binghamton.edu/fridrich/history.html#last Page 1 of 6 20 years of speedcubing 11/25/11 10:52 AM

the cube with them back to France. That meant that I could finally start working on my system! During the Summer, I persuaded my parents to visit Hungary, where I bought three more cubes. It was still a challenge to get the cubes because they were not available in stores. I bought the cubes from an old lady who was selling magazines and souvenirs in the street. When I mentioned "Buvos Kocka" to her, she smiled, quickly looked left and right and handed it to me in a brown lunch bag, put her index finger across her mouth, and said "Shhh, one hundred and fifty Forints". I know all this sounds funny now, especially to those from Western countries where it was a no- brainer to buy the cube. But this is how I really started.

At first, I was using the layer-by-layer system that I learned from a Czech magazine. It was actually already quite advanced. First the first layer, then the four middle edges (just one algorithm), then flipping the edges, moving the edges, flipping the corners, moving the corners. These were my basic algorithms that I started using and I quickly got my average to about 1 minute. It was September 1981, three months after I started playing with the cube.

My last year of high school was a strange and exciting time. A kid solving the cube in public was a head-turner. Cube was a good conversation starter, although later it was termed by many as a conversation killer . It was normal for two cubers who met on a bus to start the conversation without looking at each others eyes, saying: "How do you this?" "Hmmm, could you do it slowly?" "Thanks". This is how we built our first primitive systems. I had one schoolmate from high school who had the same disease - an unconditional love for the Rubik's cube. His name was Ludek Marek. He was using the same system as me, but for some reason he was always trailing about 20 seconds behind me. He once noted while I was solving the cube pointing to my cube: "Oh, I like this "T" pattern, because when you turn the edges, the whole last layer will actually flip correctly." It was the shortest 6-move that influences only the last layer - the move that perhaps all cubers know. And that sentence stuck in my mind. It was the germ that later blossomed into the current system. I realized that in the system I was using it was possible to first flip the edges, then the corners, then position edges and position corners. This is because the moves commuted. So, what if I had an algorithm for all flipping patterns and all permutations? Then I could solve the last layer always in just two algorithms. Also, the number of patterns was not that big and they were easy to recognize fast. But where to get the http://ws2.binghamton.edu/fridrich/history.html#last Page 2 of 6 20 years of speedcubing 11/25/11 10:52 AM

algorithms? I already knew some portion of them and I gradually started adding more. Whenever I encountered an orientation that I did not know, I did it the old way - flip edges and then flip corners. And whenever I encountered a permutation for which I did not have an algorithm, I would combine the permutation from the algorithms I already knew. I began improving very steadily as I improved my system and my ability to recognize positions quickly. In December 1981, 6 months since time zero, I was averaging about 35. Occasionally, I would read an article about a student from Great Britan who solved the cube in 28 seconds, then about a guy from USA who did it in 24, etc. I was always chasing the world, trying to catch up. I often felt like it was not possible to squeeze my times anymore, as if I was already at the limits of what I can do with my system, but nevertheless, with time, I was able to get to those magic numbers I previously read in the newspaper. At that time, I was getting ready for my final examination to finish my high school. I was combining test preparation with cubing. When I was messing up the cube, I was staring into the text, learning the subject, then pausing for a while and solving the cube. And I could go like this for hours and hours. Surprisingly, during the late Summer and Fall 1981, the cubes became finally available in Czech Republic as well. The cube crazy has officially began. Local championships popped up at high schools and universities. I always participated in them often with my friend Ludek. Both of us always left the competition far behind. There was nobody I knew with whom I could compete. I was thus chasing the clock and the world.

In the Winter 1981/82, the Czech magazine Mlady Svet called for a national championship and people started submitting their times. In February 1982, the magazine printed a preliminary table showing the best ten times submitted. And I could not believe my eyes to see my name on the first place. I also noticed the name Mirek Goljan who was literallily "breathing on my back". That gave me more energy for my practicing and I went into the national championship on May 11, 1982 averaging about 25 seconds with my personal best of 18. Ten months from time zero.

I won the semifinals and 5 best advanced into the finals. Among them, Mirek Goljan and my friend Ludek Marek. The finals were in front of TV cameras. We were allowed to use our own cubes. The best-of-three time determined the winner. We all solved the cube at the same time. I won the first and second rounds and Mirek won the last third round. My second time of 23.55 got me the first place, Mirek was the second, trailing about 2 seconds, and http://ws2.binghamton.edu/fridrich/history.html#last Page 3 of 6 20 years of speedcubing 11/25/11 10:52 AM

Ludek ended on the third place. The first prize was a plane ticket to Budapest to the first World Championship.

I became a "celebrity" for a few weeks receiving a lot of letters all asking for one thing - the description of my system. The letters actually did not have my proper address, just the name and city, no street address or zip code. They all were delivered. I decided to publish my system in Mlady Svet. It contained all algorithms for permutations and orientations and a few moves for the F2L. Most people were disappointed to learn that the method is actually quite "complex" requiring a lot of practicing and memorization. Most expected a simple trick that one can explain in a few minutes. What did you say about the free lunch? I remember one really funny story that happened to me on a train when I commuted to college from my home town. A guy was sitting next to me playing with the cube. I asked him about his system. He said: "I am using the Fridrich method." I asked with a surprise in my voice: "You actually memorized ALL algorithms?" His answer was: "No, that's too much. I know only some of them." I replied with: "Well, you need to memorize all of them otherwise you are not really utilizing its strength." He looked at me frawning and said with his mouth half open: "Yeah, so what's your system?" I answered with a big smile: "I use the Fridrich method, too, because I am Fridrich." He did not blink an eye, did not say anything and handed me his messed-up cube. I solved the cube in about 20 seconds to prove my words and we both laughed at the coincidence.

I was acepted to college and still kept on improving. Later in 1982, I changed my F2L system to the current system. Before, I would do the first layer and then insert two cubies from the last layer into the middle layer. I developed the algorithms and also algorithms that moved / flipped the cubies in the middle layer. When I switched to the current system for the F2L, I instantly improved by several seconds and got my average to around 20 (15 months from time zero). By 1983, I was consistently averaging 17 seconds. I knew three more cubers capable of achieving sub-20 averages consistently. We practiced together. As the cube rage cooled down, I stopped working on my system. The second Czech Championship took place in March 1983. Robert Pergl won all three rounds (if I remember correctly) with a best of 17.04. He was using basically my system but he knew more than 600 algorithms (I was actively using about 120-150) and one could say that he was using a "multi-system". From time to time, he was able to solve the last layer in just one algorithm, perhaps due to preparing the LL a little http://ws2.binghamton.edu/fridrich/history.html#last Page 4 of 6 20 years of speedcubing 11/25/11 10:52 AM

before finishing the F2L. And he stayed cool and psychologically stable during the whole event. Psyche is a very important factor in championships. There is little value in being able to solve the cube in 16 seconds on average if the nerves slow you down to 20 during the competition. You can't win a big event unless you work on the psychological factor as well. And Robert indeed was consulting with a psychologist, preparing very carefully for the whole year. What can I say - it paid off.

I would dare to say that nothing important happened in speed cubing and cubing in general over the next decade. Then, in 1992 Herbert Kociemba developed a computer algorithm with a performance very close to the God's Algorithm (the shortest moves from any position). It was, in my opinion, the biggest event in cubing in general. Suddenly, we could obtain the shortest moves for any position and any pattern. Surprisingly, Kociemba's algorithm always seemed to find a solution within 20 face moves. The famous cube-in- cube pattern turned out to have an elegant short solution L F L D'B D L! F! D'F'R U'R'F! D as we suspected for a long time but never found it. Progress has been made in identifying the farthest positions on the cube (superflip and supertwist). To give you an idea how revolutionary Kociemba's discovery was, the previous best computer solution was always able to solve the cube within 38 moves, but could not guarantee better (Thistlewaite's algorithm). Even though Kociemba's algorithm did not provide a proof that the diameter of the cube group is indeed 20 in face counting, it has been an impressive piece of work, indeed. As the computer speed and memory increased, optimal solvers came to life, and suddenly, to my opinion, the cube lost much of its enigma.

I put my system in electronic form on the Internet in January 1997 after I had discussions with Mike Pugh on the Cube Lovers mailing list (one of the oldest mailing lists ever, established in 1980). He persuaded me that making my system available in electronic form would be useful for other cubers. I made copies of my old, now yellowish, notebook and he made those small pictures you now see on my pages. I included the patter and set up the site. I never put a counter on my page, so little did I know how popular the system became. Actually, to be completely honest, I was convinced that nobody in their right mind will have the energy and will to learn the system in its entirety. I thought that speedcubing was inactive and not popular enough for anybody to have the motivation to go through the pain of memorizing the algorithms. I know now how wrong I was. One should never http://ws2.binghamton.edu/fridrich/history.html#last Page 5 of 6 20 years of speedcubing 11/25/11 10:52 AM

underestimate the power of the cube. I still admire those of you who entered the speedcubing now. Back in 1981, the cube was mysterious. We did not have computers powerful enough to develop the shortest moves for us. We did not know if those algorithms we found by trial and error were the best or shortest. The unknown and unanswered questions were an important ingredient for many cubers. They were the engine that powered us forward. I do not intend to sound as an old lady complaining while recalling the old good days, but I am trying to convey what most of us, if not all, felt as we were trying to uncover the curtain of secrecy of the cube.

At the end of 1996, I sent a postcard to Mirek Goljan and I typed the 14- move algorithm for cube-in-cube and nothing else on it. Mirek and I have not seen each other for at least 12 years and I was already pursuing my PhD in the US by that time. We got in touch again via phone and later via e-mail. In 1997 I visited Czech Republic and after almost 14 years, we started cubing together, admiring Kociemba's algorithm, and sharing our personal stories. Mirek later joined SUNY Binghamton and pursued his PhD degree in the same field as me - steganography and digital watermarking. We became professional colleagues and today we work together on puzzles of data hiding and discovering them in digital images. After 14 years, our journeys joined again - two top Czech speed cubers uncovering the secrets of images.

That's all folks. Thanks for reading!

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Video page with finger tricks

On this page, you will find a few video clips of me solving the cube. You will need the (free) DivX codec, to view the movies. Also, if you want to view them frame by frame, I was told that you need VirtualDub under the GNU license. I have also put some of my favorite finger tricks because they are a lot of fun. In "Stick", I reach 10 moves per second for three seconds. Overall, it was a lot of fun to watch myself on the screen. It is a completely different experience!

This was my first solve in 16.74 sec. You can tell that I am still stiff in it because of the delays between algorithms. The time was not that bad considering my rough cubing style.

The F2L were in 9 seconds, which was not bad, but then I got stuck in the permutation algorithm. Yeah, I need to work on my LL permutations, I know ... but the final time was 15.98 sec.

OK, I will be critical to myself again. This one was in 16.48 sec, again with lots of delays between algorithms. I think I should rely less on speed and go for a more reliable approach that is a bit slower but more smooth without that many delays. Isn't that what I advise everybody? :)

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This one was really really weird. 14.33 sec. - a blooper was cancelled out with a lucky permutation. I inserted two corner-edge pieces in the F2L wrong and I had to switch them. Then I got an orientation that I like (four corners) and a trivial permutation. Without being lucky, I would have gotten probably around 17.something.

My favorite finger trick - the Stick named aptly after shifting gears in a car with manual transmission. No, I do not drive that fast :) By counting frames, I determined 30 moves in 89 frames, which translates to 10 moves per second sustained for 3 seconds. I know, this is just a repetitive move, so it is easier to achieve that speed. It will be very hard to beat Lars Petrus' Sune at 7 moves in 0.7 seconds. That looks like magic, indeed!

Just a finger frenzy using a repetitive slice move.

I use those four moves when finishing the permutation algorithm V (the second on my permutations page).

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Hints for speed cubing

Customizing algorithms

It is very important to customize each algorithm for your hands. Some of us are right handed, some left handed, some may prefer algorithms which use only 2 or 3 faces so that alternate twisting from left hand to right hand is avoided. Sometimes, it may be wise to perform an algorithm with the cube turned upside down, or twisted by 180 degrees. This adjustment must be done by each individual separately because everyone may have different views of which algorithms are user friendly and which are not. This takes a lot of time, but it may cut an important chunk from your total time. Multiple algorithms

As you may notice, some positions in the last layer have several algorithms associated with them. I alternate between them to minimize turning the cube as a whole, thus cutting on time. Finger shortcuts

Most speed cubists have also developed special sequences ("shortcuts or macros") of two to four moves which can be performed astonishingly fast by pushing the faces with your fingers. Yes, it does require some dexterity. On my video page you can watch me solve the cube a few times. I also perform some finger tricks. Move algorithms to your subconsciousness

It is also important that your brain automatizes the algorithms into inseparable units - elementary actions, because then you will not have to think about individual moves. The individual moves will be performed "by your hands" rather than making your brain busy. At this stage, one can afford to think more about the next step rather than about the algorithm which is being performed. It is done for you automatically by your subconsciousness! I noticed that this automatization goes that far that if I am interrupted while performing some longer algorithm, I will not be able to finish it! In a sense, I do not know the sequence of moves and perceive the algorithm as one unit. This may sometimes create comical situatioins when somebody asks you about a specific move, and you will not able to show it slowly - and will get stuck after several moves having to start over again to see the remainder of the algorithm.

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No delays between algorithms

Another thing which is very important is to cut on delays between consecutive algorithms. One should minimize the decision time to almost zero. This issue is strongly connected with another one - the question of twisting speed. Faster twisting does not have to mean shorter times

Dogma: One needs to be especially dextered to be able to solve the cube that fast (in 17 seconds). I would be lying to say that some dexterity is not important, but I insist that an average person possesses the necessary dexterity to solve the cube in really short times. I believe that almost everybody can achieve the twisting speed of 3 twists per second. Remember, all you are required to do is to learn a finite set of algorithms perform quickly. This relates to the important issue of adjusting algorithms for your hands. So why is it possible that faster twisting speed may bring you longer times? By performing the moves really fast, one deprives him/herself of the [important] knowledge of what is actually happening to the cube. After performing an algorithm, one is then suddenly thrown into a new position and needs some time to decide which move to choose next. If you had turned the cube just little slower, you could actually see what is happening to the cube, and choose the best next move during the last couple of moves of the previous algorithm. If you compare the times: fast turns + delay between moves and slow turns + shorter delays, you will find out that the second summation may be shorter! Another argument for the second alternative is that it is very hard to turn the cube really fast, and one often encounters "stuck" cubicles, or breaks the cube to its atoms. This can slow you down as well as frustrate. Preparing the cube for record times.

I have heard people recommend a variety of different lubricants for the cube. Among others, sillicon oil, graphite, and soap were mentioned. From my experience, sillicon oil worked best. Be careful before using other lubricants because some of them may be pretty aggressive and speed up the aging process of your cube. Intense twisting causes a fine dust to develope inside the cube. Some cubists say that this kind of natural lubricant is the best one. I recommend to grease the cube because a lubricated cube will turn easier and you will be able to "cut corners" while speed cubing. But be aware of the fact that putting lubricant into a cube will make the cube more vulnerable to an accidental dismemberment. Hard work

I would like to end with a couple of more remarks on the cube. First, the secret of

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achieving amazingly short times is not just the algorithms themselves. After all, a system will never solve the cube. Humans do! Probably the most important factor is dedication and a lot of practicing. As you may notice, some positions in the last layer have several algorithms associated with them. I alternate between them to minimize turning the cube as a whole, thus cutting on time.

So, what is the best system for speed cubing? I do not think that there is such thing as the best system. One system may better fit one person, other system may be more natural for somebody else. I believe that any system which is worked out into sufficient perfection is good. We should not be comparing systems but cubists. Those certainly are comparable. What are the limits of speed cubing?

Any algorithmic set which can be performed by a human must be limited to a couple of hundreds at most thousands of algorithms. These algorithms need to be performed in a fast manner without too much thinking. This puts limits on the amount of time needed to solve the cube. If there was a hypothetical person who could see the shortest or the almost shortest algorithm right away in the beginning (which is quite improbable), he or she would need about 2 seconds, provided the farthest position is around 20 face moves at the twist rate of 10 moves per second. Since the assumption for this estimate will probably be unrealistic for many years to come, I estimate the limit for speed cubing at 5 seconds (the average time). One should totally abandon the concept of a record time since it has very little informational value. If somebody messes up the cube carelessly, one can take advantage of it and solve the cube in a few seconds. Therefore, for comparing purposes, I suggest to use an average of 10 consecutive times. For my system, I defined the concept of a modified record: I discarded record times whenever more than one stage was skipped during the cube solving. By skipping a stage, I mean: placing the four edges using less than 3 moves, too much luck for the four blocks (in the second layer), skipping the orientation of 8 cubicles from the last layer, skipping the permutation part in the last layer. For the first two layers, it is hard to estimate the probabilities, but the last layer can be calculated exactly. The probability that after solving the second layer, the last layer will have the correct color is 1/216, and the probablity that after orienting the cubes in the last layer one will not need to permute them, is 1/72. So, for example, if the last layer got assembled by chance right after the second layer, I discarded the time since the probability of that happening is too small: 1/(216*72). So, what is my modified record? It is 11 seconds. My best average out of ten was often 17 in 1983. I kept myself in a good shape for many years, and I can still get to an average of about 18 after all those years. Going back to 17 or lower would require a lot of effort, good cube, and a complete devotion that only a rookie can possess. So, good luck everybody and do not give up!

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Two edges, two corners transpositions

These sequences are transpositions of two corners and two edges. The shortest algorithm for a transposition of two edges and two corners has 13 quarter moves. The list below is an attempt to find all 13 and 15 quarter-move algorithms. The list was found by hands and is by no means considered to be complete. If you would like to add some new algorithms, please contact Mirek Goljan mgoljan (AT) binghamton. edu Jessica.

The two groups of the same length are sorted in alphabetical order. The alphabet is: A!s,A!,As,Aa,A,A'a,A',B!s,B!,Bs,Ba,B, B'a,B', ... ,F!s,F!,Fs,Fa,F,F'a,F'. Faces A,B,C,D,E,F represent U,R,D,L,B,F respectively in English notation and if you want you can easily transfer algorithms to this or any other notation. Every algorithm is put in its minimal form (in alphabetical order) of all its cyclic shifts, inversions, mirrorings, and whole cube rotations. Number of a sequence means a number of quarter moves, [number] means number of used faces.

moves faces name A²B²C E C'B²A F'A F (13) [5] CE1351 A²BaA'B'A D'A²B A'B' (13) [3] CE1331 A²B A B²C'F C B F'A F (13) [4] CE1341 A²B A B'A D F D'A'F'A F (13) [4] CE1342 A²B A B'A E'A B A B'A'E (13) [3] CE1332 A²B A B'A E'B E A E'B'E (13) [3] CE1336 A²B A B'A F'A E D E'A'F (13) [5] CE1352 A²B A B'A F'A'B'F B A F (13) [3] CE1333 A²B A B'A F'D C F C'D'F (13) [5] CE1353 A²B A F'A E D'A'D F A E' (13) [5] CE1357 A²B A'B'A'E B E'A'E B'E' (13) [3] CE1334 A²B A'B'A'E D A'F'A D'E' (13) [5] CE1355 AaB'A'B C'B A F B'F'A'B' (13) [4] CE1344 A B A B'A D F A'F'A F A D' (13) [4] CE1343 A B A B'A F'A E D A'F A E' (13) [5] CE1354 A B A B'A'E'A'E B E'A E B' (13) [3] CE1335 A B A B'A'E'A'E D A'F A D' (13) [5] CE1356 moves faces name A²BsA E A'E'B'A D A D'A D (15) [4] CE1501 A²BaA B²D'C'D F²D'C B (15) [5] CE1502 A²B A²B'F'B'F B F'D F D'F (15) [4] CE1503 A²B A²E A E'B'E'F'A F A'E (15) [4] CE1504 A²B A²E D'E D E²B'E'B E (15) [4] CE1505 A²B A B²A D A'B A B D'A B' (15) [3] CE1506 A²B A B²A'E'B E B A F'A F (15) [4] CE1507 A²B A B A'B'A F A'B'A B A F' (15) [3] CE1508

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A²B A B F B F'A'B²A F'A F (15) [3] CE1509 A²B A B'A B A E'B'F'B E B'F (15) [4] CE1510 A²B A B'A B F'A E A'F A E'B' (15) [4] CE1511 A²B A B'A B F'D'A'D A F A B' (15) [4] CE1512 A²B A B'A E'B E A D'E'B'E D (15) [4] CE1513 A²B A B'A F'A D E D E'A'D'F (15) [5] CE1514 A²B A B'A F'A D'A'D F D F D' (15) [4] CE1515 A²B A B'A F'A D'A'F'D F D F' (15) [4] CE1516 A²B A B'A F'D F²D'A'F A F' (15) [4] CE1517 A²B A B'A F'D F B F'D'F A B' (15) [4] CE1518 A²B A B'A F'D F D C'D'C F D' (15) [5] CE1519 A²B A B'A F'D F D'F A D'A'D (15) [4] CE1520 A²B A B'D'A B A'D A E A E'B' (15) [4] CE1521 A²B A B'D'E D F'D'E'A D A F (15) [5] CE1522 A²B A B'E F'A'D'A D A E'A F (15) [5] CE1523 A²B A B'E F'A'F A E'A B A B' (15) [4] CE1524 A²B A B'E F'D'E'A E D E'A F (15) [5] CE1525 A²B A B'F'B'F B A F'D F D'F (15) [4] CE1526 A²B A B'F'D'A D A E'A F A'E (15) [5] CE1527 A²B A D'A E A²B'A'E'A'E D (15) [4] CE1528 A²B A E A'B'A E'A E D'E D E' (15) [4] CE1529 A²B A F B F'B'F'A F'D F D'F (15) [4] CE1530 A²B A'B'A'B A'E D E'B'E D'E' (15) [4] CE1531 A²B A'B'A'B E'C'E A'E'C E B' (15) [4] CE1532 A²B A'B'A'D A'F'D F'D'F A D' (15) [4] CE1533 A²B A'B'A'D A'F'D'A E D'E'D (15) [5] CE1534 A²B A'D A B'A F A B A B'F'D' (15) [4] CE1535 A²B A'E'B C'B'E A'B'F'B'F B (15) [5] CE1536 A²B A'F'A B'A'F A'B'F'B'F B (15) [3] CE1537 A²B A'F'A B'A'F A'F B F'B'F' (15) [3] CE1538 A²B E A E'B'A F'A F B'F B F' (15) [4] CE1539 A²B E B E'B'A B'A F B'C B F' (15) [5] CE1540 A²B E D A'D'A'E'A'B E'B'E B' (15) [4] CE1541 AsB A'B'AaB'A'B E'C'B'C E (15) [4] CE1542 AsB C B'A'E'B E C'B'C E'B'E (15) [4] CE1543 A B A B'A D C'F A'F'A C F A D' (15) [5] CE1544 A B A B'A E'A B A B'F A'E A F' (15) [4] CE1545 A B A'E D'A D E'A B'F B A'B'F' (15) [5] CE1546

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Two edges - two edges transpositions

Apostrophe means 'turn counterclockwise'. Uppercase letter S means center slide, Sr (or only S) is the slide adjoining face R. Us,Rs,Ds,Ls,Bs,Fs are abbreviations for UD',RL',DU',LR',BF',FB', Ua,Ra,Da,La,Ba,Fa are abbreviations for UD,RL,DU,LR,BF,FB. The groups of the same length are sorted in alphabetical order. The alphabet is: U!s,U!,Us,Ua,U,U'a,U',R!s,R!,Rs,Ra,R,R'a,R',D!s,D!,Ds,Da,D,D'a,D', L!s,L!,Ls,La,L,L'a,L',B!s,B!,Bs,Ba,B,B'a,B',F!s,F!,Fs,Fa,F,F'a,F'. Every algorithm is put in its minimum (in alphabetical order) of all its cyclic shifts, inversions, mirrorings and cube rotations.

[P*Q] means the same as P Q P' [P;Q] means the same as P Q P'Q'

The first column each sequence denotes the number of face moves, the second column contains the number of used faces. Positions of order 2

U!sR!U!sL! 12 4 U!sRsB!sLs = [S!u;Sr] 12 6 U!R!U!R!U!R! 12 2 U!R!U!RaF!R'a 12 4 U!R!U!L!D!L! 12 4 U!RsB LsBsL Fs 12 5 U!RaBaU!B'aR'a = [U!RaBa] 12 5 U!RaB'aD!BaR'a = U![RaB'a*D!] 12 6 U!R B R'B'U!L'B'L B 12 4 UsRsUsLsDsLs 12 4 UsR UsF'DsR DsB' = SdB [Sd;B']S'cB' 12 5 UsR U R'F'DsR B U'B' = [Sd;B U B'R'] 12 5 UsR U F'R'DsB R U'B' = [Sd;B U R'B'] 12 5 UaR U R'F'U'aB'U'B L 12 6 UaR'U'R B U'aB U B'R' 12 4 UaR'B'R B U'aF L F'L' 12 6 U R U'R'B R B'U'B U R'B' 12 3 U R U'R'B L U'F'U F L'B' 12 5 U R U'L'U B R'U'L F U'F' 12 5

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U R U'B'R F'R B R'U'F R' 12 4 U!R D R'UsR'UaR'D'R F 14 4 UsR DsR B R'B'R'B'R'B R 14 4 UsR B D'B'R'DsB L D L'B' 14 5

Positions of order 4

U!R UsF'U!F'DsR 12 4 U!R U R U RaF'R'F'L' 12 4 U!R U R'B'R!U'R'U B 12 3 U!R U B R B!R'U'B'R' 12 3 U!R U B L U!L'U'B'R' 12 4 UsR U R DsB'U'B' 12 4 UsR U'R DsB'U B' 12 4 UsR DsB UsR'DsB' 12 4 UsR B R B DsL'B'L'B' 12 5 UsR B R B'DsL B'L'B' 12 5 UsR B R'B DsL'B L'B' 12 5 UsR B'R B'DsL B'L B' 12 5 U!R U R B'R'D'R!D B U'R' 14 4

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Twisting and moving corners and edges in one layer

At most four corners are not correct Corner rotations Corner triangles Corner cross-transpositions Corner parallel-transpositions Corner triangle and one corner rotation At most four edges are not correct Edge flips Edge triangles Edge cross-transpositions Edge parallel-transpositions Edge triangle and one edge flip Twisting or flipping cubes in place One corner and two adjoining edges are correct

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Corner rotations

The usual notation U, R, D, L, B, F is extended to slice and antislice moves. Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'. Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB. Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D. [P*Q] means the same as P Q P' [P;Q] means the same as P Q P'Q' Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. For example dFR'U = LF'Ud = LF'U. Character ^ represents mirroring (usually R<->L).

The numbers in the second column denote the number of quarter moves and face moves.

[R B'R'U'B'U;F] 14 L F U L U'R U L'U'F'L'F R'F' 14 [U;F'D R'D!R F] 16,14 [U;F'D L'F!L F] 16,14 R'F!U F U'L'U F'U'F!R F'L F 16,14 U'B U'B'U!BaU F'U F U!B'a 16,14 U'F!U L!D'L U L!D L!U'L F! 18,13 R(R F'D!F R'U!)!R' 18,13

Inversions of previous

D'F U F'D F L D L'U'L D'L'F' 14 (D L'B!L D'F!)! 16,12 [L'B L U B U';F!] 16,14 R F'L'FaU!B'R!F!L F'R F' 16,13 R!D!L D'R D'R D'L'UaF!U' 16,13 L'B'RaF'L'FaR'F'RaB'R'Fa 16 D'F D F!D'F L!F'U'F U L!D F! 18,14 U'(R'F'R U'R U R'F)!U 18

R'(D'F L'F'L F'D F)!R 18

U'(R'F'R F'U F U'F)!U 18 U'(F'R'F R'D R D'R)!U 18

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R F U F'U'F U F'U'F R'D'L'U'F'U L D 18 [U B D'B!U'B D;F!] 20,16 R'F'D'F D F'D'F D!R UsF'D'F Ds 18,17 [U'L B!U L!B'L;F!] 22,16

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Corner triangles

The numbers in the second column denote the number of quarter moves and face moves.

U!R!U'L'U R!U'L U' 12,9 R'D R'U!R D'R'U!R! 12,9 UaB!U'F'U B!U'F D' 12,10

D F!R'B'R F'R'B R F'D' 12,11

R'F!R D'R'D F!D'R D 12,10 L'U!F D F'U!F D'F'L 12,10

L!D!L U!L'D!L U!L 14,9

R!B'R F!R'B R F!R 12,9

R'aD'R U!R'D R U!L 12,10

L U L'F!L U'L'U F!U' 12,10 D F'L'F R!F'L F R!D' 12,10

D R!D L!D'R!D L!D! 14,9

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U L D'L'U'L D L' 8

R'D'R U'R'D R U 8

U'R'D'R U R'D R 8

L D'L'U L D L'U' 8

R'F!R'B'R F!R'B R! 12,9 L'U!R'D'R U!R'D R L 12,10

L'U'L B'L'U!L B L'U'L 12,11

L D'F!D L'D'L F!L'D 12,10

[R;F'L'F] 8

[F D F';U'] 8

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Corner cross-transpositions

The usual notation U, R, D, L, B, F is extended to slice and antislice moves. Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'. Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB. Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B,U,D. [P*Q] means the same as P Q P' [P;Q] means the same as P Q P'Q' Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. For example dFR'U = LF'Ud = LF'U. Character ^ represents mirroring (usually R<->L).

The numbers in the second column denote the number of quarter moves and face moves.

R'aF!RaUaF!U'aF! 14,11 [R'aB!Ra;F] 14,11

U!R!sD!B U!R!sD!F 18,10

R!D!R'U'R D!R!D R U R'D' 16,12

U L D'L'U'L!U!L'D L U!L! 16,12

R'F'R F'R'F R F'R'F R F'R'F!R 16,15 L'U!R U R'U'R U R'U'R U R'U L 16,15 (D'F U F D F U'F')! 16 D'F'L D!B!R'B'R B'D!L!F L D 18,14

U'R'F D!L!B'L'B L'D!F!R F U 18,14

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D'L'U'L D R'aU L U'R U 12 L D L'U L U'aR U L'U'R' 12 D'F!D R!U'R'U'L'U'L U'R' 14,12 L'F'L F'L F L!F L!F!L'F! 16,12 D'F!D!F D!F D F'D F'D'F! 16,12 D'F!UaL D!L D L'D L'U F! 16,13 D'F'aL!F L!B D F'D F'D'F! 16,13 D'L'U'L D L'U F R F'L F R'F' 14 L'F R F'L F U R'D R U'R'D'F' 14 R!L D L B R'B'R!L'D'L'F'R F 16,14 R BsL'FsR'F'R!B L B'R!F 16,14

R'D R F R!L B R D'R'B'R!L'F' 16,15

R D'B!D R'F'R D'B!D R'F 14,12

U F!U!F'LsF!RsF'U!F!U 18,13

The three-tuple associatedwith each algorithm denotes the number of quarter moves, face moves, and slice moves.

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Corner parallel-transpositions

The usual notation U, R, D, L, B, F is extended to slice and antislicemoves. Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'. Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB. Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B,U,D. [P*Q] means the same as P Q P' [P;Q] means the same as P Q P'Q' Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. For example dFR'U = LF'Ud = LF'U. Character ^ represents mirroring (usually R<->L).

The numbers in the second column denote the number of quarter moves and face moves.

UaL U'R U L'U'aL D R'D'L' 14

R'F L'B!L F'RsF R'B!R F'L 16,14

U'R'F L'F!RaUaF!D'L F'L'F! 18,15 D'L'D R'D'L D'L!D'R'D L!D'R D'R 18,16

D F!L'F!RaUaF!U'aF!R'F!D' 20,15

D'L'DsR'UsL DsR U 12 L'U'R'F R U L D R F'R'D' 12

D'L'F'L F L D RsF'L'F Ls 14

U'F D F'U F UsL D L'U'L D'L'F' 16 L'F R F'L F LsD L'U L D'L'U'F' 16 R'D'L D'L'D!F!R U'R U R!F!R 18,14

U'F!D'B!D F!U F'U'F D'B!D F'U F 20,16

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L'F!L'B'L F'D F'U'F D'F'U L'B L! 18,16

U!R!U'L'U R!U'L U!R U L'U'R'U L 20,16

U'R'D F'L!F R F'L!F U R'D'R 16,14

U'R'D'F!D R U!L'U'L!F!L'F! 18,13

U F!R B'R'F R B R'D'F U'F'D F 16,15 U F!R'D R D'F!U'R'F!R!F!R' 18,13

R'F!R F R'F!D F'D'F'R D F!D'F' 18,15

R U L'U'R'U B'L F L'B L F'U' 14 D R!U'R'U!F!U'F!R'D'L'F!L 18,13

D'L'U'F'R!F L F'R!F D L'U L 16,14

U L F L'F'(L F L'F')!U' 14 [R'D!R!F!R'*D!]F! 20,12 [R'D!L!B!R'*U!]F! 20,12

R'F!RaF!R'F!R F!R'aF!R F! 20,14

U'R U'R'U!L'U L U'L F!L'F! 16,13 [U'R U'R'U R';F!] 16,14 (R'F L F'R F L')!F! 16,15 U'R'D R U R'D'R!U'R'D R U R'D' 16,15 R'F!R U'R U R!F!D'L D'L'D!R 18,14 U'F!D'B!D F!UsF!U'B!U F!D 20,14

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D'B D!B L'U'L B'D!B L'U L B!D 18,15

R'F!R F!R U'R D R!U R'D'R F! 18,14

R U'R'F'R B U L U!L'U B'R'F U 16,15 R F R'D'L F!L!F'L!F'L'F D F! 18,14 D'L'U'L D L'U LsD'R U'R'D R U 16

R'F'D'R'aF'R'D R D'F RaF D R 16

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Corner triangle and one corner rotation

The numbers in the second column denote the number of quarter moves and face moves.

U!B'D B!U F'U'B'U FsD'B U 16,14

L B R'FsL B'L'F'L B!R B'L! 16,14

R'D'R U'R'UaL'U'RaD'L'U L D 16 R'F'D F'U'F D'BsU F'U'B'U F'R 16 R'D R D L'U'L D'R'D'R D'L'U L D 16

U'R'U L'U'R U!L'D!L U'L'D!L! 18,14

D'F!D F!D'F!L D R'D!L'D'R D' 18,14

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D'F!D F!D'F D F'U F'D'F U'F D F 18,16

R'D R'U'R!D'R U F!R'F!R F!R' 18,14

U F D'F U'F'D F'U F U'F!U F!U'F 18,16

D R D'L D R'D!L'U L D L'U' 14,13

L'U'R U L U'R!D'R U R'D R 14,13

L D'L U!L!D R'D'L D RaU!L! 18,14

U!B U'F!U B'U'F D'F U'F'D F 16,14 U F'D'L F'L'B'L F L'FaU'F'D F 16

D R D'L D R'UsL'F!L U'L'U F!U' 18,16

U F'D'F U'F D'B'D F!D'B D!F 16,14 U F'D'FaL'F L B'L'F'L U'F'D F 16

L'F!L U'L'U F!U'LsD'R U R'D R 18,16

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U'F D F'U!B!U'F U B!U'aF' 16,13 U'aB!U F U'B!U!F'D F U'F' 16,13 U'F'U!B'D B U B'UsF U'B U! 16,14 R'F!RaF!R'F'R F!R'aF!R F' 18,14

R'F R F D'F!D F'R'F R D'F!D F! 18,15

U'R!D'R U L'U'R'D R!UsL D 16,14 L'U'R U L U'R'U!L U'R U L'U'R' 16,15 L'U B U'F!D'F U F'D FsU'F L 16,15 R B L'U L B'R'U F'U!R'F'R F U 16,15

R'D F!D!L'D R D'L D!R'F!R D' 18,14

R U LsD!R D'L'U'L D R'D!L' 16,14 D'L'U'L D L'U L!D'L'U L D L'U' 16,15 U F L'FsR F'L F R'F!L'B L U' 16,15 L F D F'D'L!F'L D'B'U L U'B D 16,15

R'D F!D'R!U R'D R U'R!F!R D' 18,14

U L U'R U L'U'R!D R U'R'D'R U 16,15 R'D'F'UaL'U'R U L!D F D'L'U' 16,15 L'U L U'F!L!D R'D R D!L!F! 18,13

R!U!R D!R'U!R D L D R D'L'D 18,14

D'L D R'D!L'D'R'D'R!U F!U' 16,13

U'R'D R'D'R'U R'F!R'F!R F! 16,13 U'R'D R'D'R'U L'U!R'U!L F! 16,13 U F!U'F!U'R U'L'U'L U'R'F! 16,13

D'F U F'D F U'R'F L F'R F L'F! 16,14

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L D R'D'L'D R D!L'U'L D L'U L 16,15 L'U'R'F R U!L D L'U'RaF'R'D' 16,15 D!L!U L U'L D!F!R'D R D'F! 18,13

R U'R'D R U R D L!D'R!D L!D! 18,14

U'F'U F U L'U!F'U F U'F'U F L 16,15 U F L F'L'F L F'L!U'L F L F'L' 16,15 D R F R'F L D!B'L!F L!B L'F D' 18,15

R'F!R F U R'F!R F!R U'R!F R F! 20,15

L'F!L D!R'D'R'U'R!D'R U R' 16,13

http://ws2.binghamton.edu/fridrich/L1/ctr.htm Page 4 of 4 Edge flips 11/25/11 10:55 AM

Edge flips

The numbers in the second column denote the number of quarter moves and face moves.

R U'B L F R U R'U F'L'B'R'U' 14 R'D'L'U'F'U F'L D R U F U'F 14 U F U!B'R'F'R FaU!F!U'F 16,13 U'F'aR!F!R F'R'F'R!B U F 16,13 U'aF!U!F U'F'U'F!D R U R' 16,13 L!U!L U'L'U'L!D F U F'U'a 16,13 UsF U'RsB U'FsR'F'LsD R 16,16 S!F S F!S'F S'F S'F!S F S' 24,20 U'F R'U F'LsD F'R D'F Rs 14 RsU'F R'U F'LsD F'R D'F 14 U'F'U F'L D R U F U'F R'D'L' 14 U'B'R'F'R FaU!F!U'F U F U' 16,14 RsU L DsF D'LsB D'FsL'F' 16 [S B S'B S;F!] 20,18 S F S F S F!S'F S'F S'F! 20,18 R F U F'L F!L'U'F'R'F D'F!D 16,14* D L U(R Sf)!(R Sf)!U'L'D' 18 U!LsF!R!sU'R!sF!RsU!F 22,14 U!LsF!R!sU'F!RsU!R!sF 22,14 R!U!R!F!RsU LsF!R!U!R!F 22,14 [S'F!S F!S';F] 22,18

* -these algorithms have been added from 'Goodbook' supplied by Razoux Schulz

http://ws2.binghamton.edu/fridrich/L1/ef.htm Page 1 of 2 Edge triangles 11/25/11 10:56 AM

Edge triangles

The numbers in the second column denote the number of quarter moves and face moves.

U!F RsU!LsF U! 12,9 R F U F'U'R'aU'F'U F L 12

S!F'S F!S'F'S! 16,11

[R U'R';Sd] 10 [Su;R F'R'] 10 [Su;R'F'R ] 10 R F R'D'L'U'F'U L D 10

F'S'F'S F'S'F S F S'F S 18

L D R F!R'D'L'U'F!U 12,10 U'R'U!B!D!L'D!B!U' 14,9

S F S'F!S F S' 12,11

[Su;L'U L] 10 [L F L';Sd] 10 [L'F L ;Sd] 10 R U F'U'R'D'L'F L D 10

http://ws2.binghamton.edu/fridrich/L1/et.htm Page 1 of 2 Edge triangles 11/25/11 10:56 AM

S'F S F S'F S F'S'F'S F' 18

http://ws2.binghamton.edu/fridrich/L1/et.htm Page 2 of 2 Edge cross-transpositions 11/25/11 10:56 AM

Edge cross-transpositions

The numbers in the second column denote the number of quarter moves and face moves.

R'aF!RaUaF!U'a 12,10 [U!S!U!F] 18,10

S!F S!F!S!F S! 20,11

R'F'D'F!D R F U F!U' 12,10 U F L F!L'U'F'R'F!R 12,10 [S'B S B S';F] 18

S'F S F S'F S F S'F S F' 18

R U'R'F L'F R F'U'L F'U F R' 14 U F'R FaU!F'aR!F!R F U' 16,13 U S'U F S!F!S!F U'S U' 20,15

[S'B S B S';F] 18

U!D'B L'B'D U!R'D'F D R 14,12 R'F R'D'F U'F D F'R'U F'R! 14,13 RsB U B'LsF'DsR U R'D' 14 RsF U F'U'F'U'F'U F LsF 14 R'F'D R'D'R F R D'L'F'L F D 14 R'D'L'F L F'D R D'L'F'L F D 14 http://ws2.binghamton.edu/fridrich/L1/ect.htm Page 1 of 2 Edge cross-transpositions 11/25/11 10:56 AM

R'F'D'R U F'L'F L U'R'F D R 14 R'F'D'L F D'L'D L F'L'F D R 14 [U!S'U!S!F] 22,14 [S'F!S'F!S';F] 22,18

http://ws2.binghamton.edu/fridrich/L1/ect.htm Page 2 of 2 Edge parallel-transpositions 11/25/11 10:56 AM

Edge parallel-transpositions

The numbers in the second column denote the number of quarter moves and face moves.

R'F'R'D R D'F R D'F'L'F L D 14 R'D'F L'F'L D R D'F'L'F L D 14 D'F'D R!F U F'U'R!F'D'F!D 16,13 R!F'R F R U!R'B'R B U!F'R 16,13 S!F U!S!U!S!F'S! 22,12 S!F S F!S!F!S F'S! 22,14 S F S!F S!F S F!S!F' 22,14 R'F'D'F D R!F U F'U'R' 12,11 R!F U'R'D B R'B'R U R'D'R' 14,13

S F'S'F S F!S'F'S F'S'F S F!S'F 24,22

R'F'D R'D'R F R D F R F'R'D' 14 R'D'L'F L F'D R D F R F'R'D' 14 R'D'L'F L D F!R U L F'L'U'F! 16,14 R'D'L'F!L D F R F!R'D'F'D R 16,14 R'D!L D L'D R D F!D'F'D F'D' 16,14 L'F L!R'D L'D!L D'R L!F'L F! 18,14 L'D!U'R F D'R!D F'R'U D!L F! 18,14 [S'F S F S';F!] 20,18 U R U LsF'L'F'RsU LsU' 14 R F U'R U R D R D L D L'F D 14

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R F'R'D'F!D F!R F!R'D'F'D 16,13 U!S'F S F S'F'S F'U! 16,14 S'F S F'S'F S F S'F!S 18,17

F S F S'F S F'S'F S F S' 18

U'aL FsLsF RsU!BsUa 16,15 R'aU'R!F!R!F!R!F!U Ra 18,12 (U R'S!fR U'F!)! 20,14

S F'S'F!S F'S!F S F!S'F S 24,20

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Edge triangle and one edge flip

The numbers in the second column denote the number of quarter moves and face moves.

D'F'D L B D!B'D'L'F D!F!D' 16,13* R F!R'F D'L'U'R'F'R F!U L D 16,14 R F R'D'L'F'L DaL D F'D'F L'U' 16 U'R'F R U L F'L'U F L'U L U'F'U' 16 RsF'U!F!U!F!U!R'F'LsD R D' 20,15 [S'F S F'S'F!S ;F] 24,22 D L D LsB'L'B D'RsF D! 14,13 F L B D'F D'F'D'F D!B'aL' 14,13* U L F'D F D'L'U'R'F'D'F D R 14 U F U L'U'L F'U'R'F'D'F D R 14

F S'F'S'F!S F'S!F'S' 18,15

D'F!D!F R'D'B'D!B R D F'D' 16,13* D R U F!L F'L'U'R'D'F L'F!L 16,14 U'R'F D'F'D R UaR F'R'D'L'F L 16 U'F'U'R U R'F U R'F'R U L F L'U' 16 D'L D RsF'L'U!F!U!F!U!F'Ls 20,15 [S F!S'F'S F S';F] 24,22 L F D'L D L'F'L'U'R'F'R F U 14 L D R F'R'F D'L'U'R'F'R F U 14

http://ws2.binghamton.edu/fridrich/L1/etr.htm Page 1 of 2 Edge triangle and one edge flip 11/25/11 10:57 AM

S F'S'F'S F'S'F'S F!S'F!S F'S'F 26,24

* -these algorithms have been added from 'Goodbook' supplied by Razoux Schulz

http://ws2.binghamton.edu/fridrich/L1/etr.htm Page 2 of 2 Twisting or flipping cubes in place 11/25/11 10:57 AM

Twisting or flipping cubes in place

The numbers in the second column denote the number of quarter moves and face moves.

This is a set of algorithms for solving positions in the last layer when all the corners and edges are in place but they are to be twisted or fliped.

Positions which appear in sections dealing solely with corners and edges are not included here.

R'D!L D L'F D R U!L'U'L F'U' 16,14 U L!F!R'F L'F'RaF!L!F'U'F 18,14 R F R'F'U!R!D R'U R U'aR!U! 18,14 U'F'R'F R!B U!L'U L U B'R'U F! 18,15 D'F'D F!D'F!R'F L'F'RaF D F! 18,15 L F!L'F!D'F U'F'UaF L F!L'F' 18,15 U F L'U L U!R'D'F'L D'L'D!R 16,14 R F U!F!U'aF U F'D F!U!R'F' 18,14 U!R!UaR'U'R D'R!U!F R F'R' 18,14 U'F'R'aF R F'L F!U F!U'F U F! 18,15 D'L B D'R'D'R D!B'L!F'L F D F! 18,15 D F!D'F'L'aF L F'R F!D F!D'F 18,15 D R!U!R'B'R B U!R!D'R U'R'U 18,14 U'F!U!FaU'F'U B'U!F!R U R' 18,14 U'F R D'F D F!U'B'R'D B'D'B!U! 18,15 U'BsU F'aU!B U B'U!F!R U R' 18,15 R'F D'R'aF'D'F D RaD!R'D'R!F' 18,16 U'R U R'D R!U!B'R'B R U!R!D' 18,14 R U'R'F!U!B U'F U F'aU!F!U 18,14

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U!B!D B D'R B U F!D'F'D R'F'U 18,15 R U'R'F!U!B U'B'U!FaU'FsU 18,15 U!R U R!U'aR'F'R F UaR F'U F 18,16 U L'U'L F'U'R'D!L D L'F D R U 16,15

U!R U R!F R D'F'D!R'D'R!F'R'U F 20,16

U'R'D'F'L D'L'D!R U F L'U L U' 16,15

R'D F D!L D L!F L D'F'D!R'D'R!F 20,16

U'F!L'F R'F'RaF U F!U'F'U F! 18,15

L'F!R F R'F!L!D F D'F'RsF R' 18,15

U'F U L D F D'F'D F'D'L'U'F'U F 16 U F R'U'R F!R'F'D'F UaR F!U'F 18,16 D'F'L'F L F'L D'L'D L D'L'D L'F L D 18 R U!B L!F L'F'U'F!U L'B'U R U R! 20,16 R U'R'U'B L U'F!U F L F'L!B'U! 18,15 U'R'aF'R F U F!U'L U F'L'F'L F! 18,16 U L F L'F'DsF U'aF!D F D'F!U 18,16 U!B L!F L'F'U'F!U L'B'U R U R' 18,15 R'F R F D'R'D F!D'F'L'F LaD F! 18,16 U'F!D F'D'F!UaF'UsF L F'L'U' 18,16 U L F L'F'L F'L'U'R'F'R F R'F R 16 R F!R!F'D'F D R'B R'F'R B'R!F' 18,15 R F!R!F'R F!L F'R F L'F U'R'U 18,15 U!L'U'L!D R F U'R U R!D'L'F'U' 18,15 D!R!U!R'B'R B U!R!D'R U'R'Us 20,15 R'F'R F'R'F R U L F L'F L F'L'U' 16 U!B U'F U B'U R'F'R F U!F!U'F 18,15 U'R U F'L F'R'F L'F!R'F R!F!R' 18,15 U F L D R!U'R'U F'R'D'L!U L U! 18,15 DsR U R'D R!U!B'R'B R U!R!D! 20,15 U'F'U!L'U'L!F'L'F! 12,9

U'R U'R'U!F'L F'L'F! 12,10

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D'F!R'F'R!U'R'U!F'Ds 14,11 R'D'F!R F U F'U'R'D F R F' 14,13 R'D'F'D F R D'L'F'L F D F! 14,13 UsF U!R U R!F R F!D 14,11 U'F'R'U L F L'F'U'F!R U F 14,13 U'F'R'F R U L'F'U'F U L F! 14,13 L'B'U F'D R U R'D'F U'B L F 14

U L F!U'F'U!L'U'L!F'L!U' 16,12

U'B'R F'L D R'D'L'F R'B U F' 14

U L!F L!U L U!F U F!L'U' 16,12

R!L B'D!R D F D'F'R'B D R D'Rs 18,16

D F R UaF R F'R'U'aF'R'D' 14

U L F UaL F L'F'U'aL'F'U' 14

U F U!F'L F R'L'F R F'U!F U F! 18,15*

U F U!R U R!F R!F R!D R D!F D 20,15

L'U'R'F R U L F D F'D'F D F!D' 16,15

D F!D'F'D F D'F'L'U'R'F'R U L 16,15

U L F L'F'U'aL'F'L F D 12

D'F'L'F L UaF L F'L'U' 12

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D'F'D F'D'R'aF'D'F D RaF!D 16,15

U'R'U L'U'R U'R!B!L'D L B!R!U!L F' 22,17

D'L!U!B!D'R'D B!U!L U'L D L'U L F 22,17

L'F'L!D'L'D!F'UsF U!R U R!F R 20,16

D'L!U!B!D'R B U B'D B U'B U!L!D F' 22,17

L'F'U'F U L U L F L'F'U'F! 14,13

U L F'D F D'F'L'U'F'R'F!R F! 16,14

(R'D'F D R U F'U')! 16

D B R'F L'U'R!U L F'R B'D'F! 16,14

RsB U B'R'B L F'L'B'L U'R U R'F! 18,17 D F'U'F D'F'D B R B'UsF'R'D R D'F' 18 (U'F'U!L'U'L!F'L')! 20,16

* -these algorithms have been added from 'Goodbook' supplied by Razoux Schulz

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One corner and two adjoining edges are correct

The numbers in the second column denote the number of quarter moves and face moves.

This is a set of algorithms for solving positions in the last layer when one corner and two adjoining edges are correct.

Some of these positions are already included in sections dealing with corners and edges and in the section on twisting and flipping cubes in place. They are again included here for the completeness.

U!R!U'L'U R!U'L U' 12,9

R'D R'U!R D'R'U!R! 12,9

U!B'U'B U'L!D F'D'L! 13,10

L F'R F!L'F L F!R'aF 13,11

R'U'R'U R U!B U B'U F R F'R 15,14

R B'R'U'B'U F U'B U R B R'F' 14

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U'F!U L!D'L U L!D L!U'L F! 18,13

R'F!R D'R'D F!D'R D 12,10

L!D!L U!L'D!L U!L 14,9

R!B'R F!R'B R F!R 12,9

L'B D!R'D R D B'L'D F D'L! 15,13 L'F L F'D R'D R D!L D'L'D F 15,14

L!D F'D'F'D F L F L'D'L F'L 15,14

L!B R'D!R B'L'F!L'F' 13,10 R'F!R F!D F'R'F R F D' 13,11

R'F L'F R F'L F!R'F R F 13,12

R'D'F'D F'R U L D F!D'L'U'F' 15,14

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U B'U'L'B'L F'L'B L U B U'F 14

R'D R!U'R!D'R'U R!D'F!D F!

L U L'F!L U'L'U F!U' 12,10

D R!D L!D'R!D L!D! 14,9

U L D'L'U'L D L' 8

R'D'R U'R'D R U 8

L F'R F R'aF!R F R'F! 13,11

U'R!UsF'D F L'U'R!U L F' 15,12

U'R!D'R!U L F'D'F'D F L'D 15,13

U'R U F D'F D R D'R'F'R D F'R' 15 U!F'U F U!L!D F'D'L!B'U B 17,13

U'R!B D'F!U'F U F'U F!D B'U 17,14

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D'F U F'D F L D L'U'L D'L'F' 14 D L'B!L D'F!D L'B!L D'F! 16,12

[L'B L U B U';F!] 16,14

R'F!R'B'R F!R'B R! 12,9

L D R!D'L'F U F'DsR!D'F' 15,13

L!B R'D!RsF'L B'L'F'L'F' 15,13

U'F D F D'F L!B'U B L!F U 15,13

U'R'D'R U R'D R 8

L D'L'U L D L'U' 8

L'F!L'B'R U!R'B L!F' 13,10 L'F U F U'F'L F!U F!U' 13,11

L F L'F!R F'L F R'F L'F 13,12

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R'D'F'D F D!L D L'D R F 13,12

U L D'L!B'L'B L'D L'U'F' 13,12

R F'L'FaU!B'R!F!L F'R F' 16,13

L D'F!D L'D'L F!L'D 12,10

R F'L'F R'F'L F 8

L'F L'F'D F!L F!L'D'F L! 15,12

D!F L'D'F!D F!L F'D'F D' 15,12

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U'F!U!F U!F U L'F U F U'F'L 17,14

U'aF!U!F U'F'U'F!D R U R' 16,13 U F U!B'R'F'R FaU!F!U'F 16,13

R'D'L'U'F'U F'L D R U F U'F 14

U!F'U'L'U'L!F'L!U'L U F! 16,12

U!F'U'R'U L'U'L U'F'U F R U'F 16,15

U L U L'U B!D'R D B!U 13,11

D!B R!U'R U R B'U'B R B'U D! 17,14

U'F'U!L'U'L!F'L'F! 12,9

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R U'R'U F'L U L'F L U'L' 12 R'D F D'R D F'D'R U'R'U 12 R'F D!B'D F D'B D!F'R F' 14,12 R F'U!B U'F U B'U!F R'F' 14,12 R B'R!F R F'R!B R'U'R'U 14,12 R B'R!F D'F D F'R!B R'F' 14,12 R'B R!F'U F U'F R!B'R F' 14,12 L U'F!U!F L F'L'U!F U F L' 16,13 L F R'F!R!F U F'U'R!F Rs 16,13

U R'U!F D'F'U F!D F'U R U'F' 16,14

U R'D!L'D L D'L D!R U'F! 15,12

L'D R!U R'U R U'R!D'L F! 15,12

R B!D F'L!F D'B'U!B'U'R' 15,12

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U F L'U L U!R'D'F'L D'L'D!R 16,14

R F U!F!U'aF U F'D F!U!R'F' 18,14

R'F L'D'L F L'D L F'R F' 12 R'F D'L F L'D L F'RsF' 12 R'F'U F!D'F D F!U'F R F' 14,12 R'F!L F'R F L'F!R U'R'U 14,12 R'F!L F'D'F D F L'F!R F' 14,12 L'U!R B'R B R'U!L U'R'U 14,12 L'U!R U R'U!L U'F'U'F U 14,12 L F!L'F'U'F'U F L F!L'F' 14,12 L'F'L F R F'U L'F L U'F R'F' 14 L'F'L UaF U'R F R'UsF'U' 14 U!B U!R U'R'B'R U R F R!F' 16,13

L D R F U F'U'F'R'F!D'F!L'F' 16,14

R!F R F'R'F!D'F'D F'U'R'U R! 17,14

R F U F'R'U F L F'U'R U L'U!R'F' 17,16

R B'U R U R'U'B U!R U!R! 15,12

R'F!R F'U'B'D R'D'B U F' 13,12 http://ws2.binghamton.edu/fridrich/L1/ece.htm Page 8 of 11 One corner and two adjoining edges are correct 11/25/11 10:57 AM

R U'R'F!U!B U'F U F'aU!F!U 18,14

U F'L D'F'D L'F U'F'D'F D F 14 U F DsR F'R'U F'U'aL'F L 14 U!F'U'L'U'B U L U'L!B'L!F 16,13

D F!R F!U F L F L'F'U'R'D'F 16,14

U R U'B'D'F R'F'D B R!F R F' 15,14

R F!D F D'F L!B'U B L!R' 15,12 D R!U'aR U R'D R!U'R Us 15,13

D R B'R U'R U R!B R'U'R Us 15,14

U'F!U!FaU'F'U B'U!F!R U R' 18,14

D R F'U F DsF D'R'D F'D! 14,13 R F'L'U'F'U B D L'D'FsL U R'F 16

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L!F R'F'D!F!D F'D'F'D!L!R 18,13

U!R!U R!B R'U'R U R B'U 15,12

U'R!U R'U'R!UaR'U'R Us 15,13

DsR U F R F'aR F R'B R'F'D' 15

R'D!L D L'F D R U!L'U'L F'U' 16,14

U L!F!R'F L'F'RaF!L!F'U'F 18,14

R F'L U L'F L U'L'F R'F' 12 RsF'L U L'F L U'F R'F' 12 D'R D F'U F U'F D'R'D F' 12

R F D'F!U F U'F!D F'R'F' 14,12

D!F D'R D F'UsF'U'F R'D' 14,13

R!F R F'U'F'aR B R'F R!U 15,13 http://ws2.binghamton.edu/fridrich/L1/ece.htm Page 10 of 11 One corner and two adjoining edges are correct 11/25/11 10:57 AM

R F R!B U F'R'F U'B'D'R D F' 15,14

U'aR D'R'B R!D R!D'B'R U D! 17,14

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Pretty Patterns

This is the largest collection of pretty patterns that is known to me. It has been compiled by Mirek Goljan, who is also the author of most of the moves. Significant contributions were made by Peter Nanasy, Michael Reid, Mark Longridge, Hans Kloosterman, and others. Contents:

Notation and terminology Regular patterns Semi-ornaments Letters on all faces

U's ornaments U's semi-ornaments Supplements of the U's (Czech check problem) T's simple patterns H's J's ornaments Supplements of the J's I's semi-ornaments K's Supplements of the K's L's

6 L's from Q2 6 L's from Q1

Regular ornaments Regular ornaments -supplements Specials Semi-regular patterns Semi-regular ornaments Semi-regular ornaments - supplements Ornaments composed from U2, U3 or U4 Semi-regular semi-ornaments

Even positions Odd positions

Cyclic patterns Three-color regular patterns Three-color regular ornaments More-color regular patterns More-color nonregular patterns

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Flips and twists

Patterns possible by disassembly Ornaments Regular semi-ornaments Semi-regular semi-ornaments Cyclic patterns

Simple patterns One little square of opposite color Two little corner squares of opposite color

Odd simple patterns

NOTATION AND TERMINOLOGY

Here is Mirek's classification of two-color patterns which he introduced in 1982.

1. Ornaments A typical example of an ornament is "cube in cube". Colors of the faces are mixed: (U->R->F)(D->L- >B). Axis of each pattern goes through the corners URF and DLB. In order to desribe ornaments in a short, symbolic, form the following notation is introduced: D is an exchange of the centers (F,U,R) (L,D,B) V are twists of the corners FUR+ and LDB- S is a permutation of the edges (FU,UR,RF) S' is a permutation of the edges (LB,BD,DL) L is permutations of the edges (FU,FR,UR) (LB,LD,BD) O is a permutation of the corners (FUL,URB,RFD) O' is a permutation of the corners (LBU,BDR,DLF) H is a permutation of the edges (UL,RB,FD) H' is a permutation of the edges (UB,RD,FL)

2. Semi-ornaments Every face has at most two colors which are mixed: (F<->R)(B<->L)(U<->D) D is an exchange of the centers (FR)(BL)(UD) W are flips of the edges FR and BL.

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O is an exchange of the edges FL and RB. R is an exchange of the corners FRU and RFD R' is an exchange of the corners BLU and LBD K is an exchange of the corners FLU and RBD K' is an exchange of the corners FLD and RBU S are permutations of the edges FU, FD, RU, RD S' are permutations of the edges BU, BD, LU, LD D is used to shorten the notation. [D] = [WORR'KK'S(14)(23)S'(14)(23)] The alphabetical order of letters is: D,W,O,R,K,S. Patterns containing all of R, R', K, K', S, S' are mostly denoted as [D...]. You should use mirroring (F<- >R) to get K rather then K' and S(1.. rather then S(3.. or S(2.. .

3. Cyclic patterns Every face has at most two colors which are mixed: (F->R->B->L)(U<->D)

4. Simple patterns Every face is a simple two-color combination of colors from opposite faces of the Cube: (U<->D)(R<- >L)(B<->F)

5. Specials Other permutations of colors, for ex. (F->L->U->B->R->D) or (F->B->L->R) (U<->D) REGULAR PATTERNS

Regular patterns are patterns which have all faces of the same look except for coloring and mirroring.

Semi-ornaments

F U L'B°L F!D'F D F L'D'L F L'D B°L 21,20 [OKS(14)S'(23)] 2 peaks U'F' ° can be replaced by the same but otherwise arbitrary exponent p117 Improve RCC U F'D'L B!L'D F D L!U'aF!D'F!L! 21,16 (K) 3.9.36 D U!F!D!sR!sB!R!sD 20,11,8 [RR'KK'] crosses D R!sDaRaF!sRaU 16,12,10 D RsFaL!F'aLsD'aL!U 16,14,12 p147 ML's 6 ARM U'R'F'U'R!sD B R'L!D!B!D!sB!U' 22,16,14 [ORR'KK'S'(12)] (K) Pattern R U'F R'B'D!R'U'BsLsD B L U'R'U'F 23,22,19 [DK] MG.1982 U'Ls RaB'D'aL'R!D!F!R F'aU'F!R!U D!L! 25,18 p68 Cherries (K) D'F!DsB!RsU RsB'R!B LsU RaB!U 24,20,16 [DOS(23)S'(14)] (R!D L!)^4 Ds 22,14,10 [RR'KK'S(1342)S'(13)] p163 4 ARM Full

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Letters on all faces

The U's - ornaments

BsU F'LsD F'U'BsR 12,12,9 U1 = [DH] P.Nánásy F'R!D'RsF RaF D B LsU'F' 16,15,13 U2 = [DL] E.V.Chacøn U B U'LsDsBsD R'UsB U'B' 16,16,12 U3 MG.1983 B R D'R FsD'RsF UsL'F U'L' 16,16,13 MG.1982 L D BsR'F L'F'DsR F R'D' 14,14,12 U4 MG.1982 R B'LsD U!BsR'L!DsF D' 16,14,11 R B!F UsL'FsU RsB'F!D' 16,14,11 L'F'aR'DsB R F LsD'L D 14,14,12 U5 MG.1982

The U's - semi-ornaments

R'aB!RaD'aL BsD'B!D FsL'D! 20,17,15 U6 [0+0] MG.1982 B LsD'LsB!RsD'LsF'aD!sFDs 22,19,13 U'LsF'UsL!B!L'BsD B! 16,13,10 U7 [4+0] MG.1983 F'L'FsR'F'D'F R BsL B L!BsD' 18,17,14 MG.1982 B F!D!BsR'BsL'F DsR'F RsB'R 20,18,14 U8 [2+2] MG.1982 R'F!LsD!B RsF L UsB'L'BsR'B 20,18,14 U9 [3+1] MG.1982 L'B'LsF'L'UsB L FsR F U!FsR! 20,18,14 U10 [1+3]

Supplements of the 'U's

Here are some positions with exactly 8 squares correct on every face. The remaining positions are discussed in the article "A Czech Check Problem" by Michael Reid, Cubism For Fun 36, Feb 1995.

LsF L'FsU F'U'BsR 12,12,9 DU1 = [H] R D'LsF'RsD LsF L' 12,12,9 F'R!D'RsF RaF D F UsL'U' 16,15,13 DU2 = [L] MG.1982 F L F'RsD R'UsB U'B' 12,12,10 DU3 /E5/ MG.1983 U L B'D BsR'FsD BsR'F L'U'=(ULB'DBsR'FL'U')! 16,16,13 MG.1982 L U B'D BsR'FsD BsR'F U'L'=(LUB'DBsR'FU'L')! 16,16,13 MG.1982 U L'DsF'UsL DsF D' 14,14,12 DU4 MG.1983 UaL'DsF'UsL DsF D! 14,13,10 DU5

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D'R'aD'RsF R F LsD'L D 14,14,12 MG.1982

The T's - simple patterns

R!U!R!U!R!FsU!Fs 16,10,8 T-ordinary MG.1982 L!D!R!D!L!FsU!Fs 16,10,8 MG.1982 L!D!B!D!F!L!B!U!B!U! 20,10 p97 6 T's A FaU!FaR!D!R!U!R!B! 18,11,10 T-orthogonal MG.1982 F!D!R!U!B!D!B!R!B!D! 20,10

The H's

R!BsR!sBsL! 12,8,5 H1. - simple pattern L!(U!F!s)!L!F!s 20,10,7 p174 6 H order 2 type 1 FaR!F!sD!sL!F'a 16,10,8 H2. - simple pattern RaDaRaD!sRaDaRa 16,14,13 R!B!U!R!L!U!F!R! 16,8 p175 6 H order 2 type 2 R'aUaR!sF!sUaRa 16,12,10 p175a R!F!sR!D F!sU!R!sD 20,11,8 H3. - semi-ornaments U!R!sD F!sR!F!sR!U 20,11,8 R!sU'F!sL!F!sL!D 18,10,7 H4. - cyclic pattern D'L!F!sL!F!sD R!sDs 20,12,8 R!sD'FsR!BsRsB!RsU 18,14,9 H5. - specials D!R!F!U'F!sD R!sF!L!D! 22,12,10

The J's - ornaments

F'R!D'RsF RsF D'R!D F'L!F D'R!D!B LsU'F' 28,23,20 DLV MG.1982 U'R B'R'U R'D!F U F U'F'D'U!BsLsD F 22,20,18 U4V MG.1997 R'U'B L'B'L!U R U'L'U L D BsLsU'L F'L!F 24,23,21 U4V MG.1997 R!F'D'F!L F'R F R'aF'D B LsDsF'U! 22,19,17 U5V MG.1997 R!F'D'F!L F'L DsBsL D'aR'B R B'U! 22,19,17 U5V MG.1997 R'B'L!U'F'U L'B R!B'L'B R DsBsL Da 22,20,18 U5V MG.1997 R!F'D'R'D F L'F'aR F R'B RaU'BsL Da 22,21,20 U5V MG.1997

Supplements of the 'J's

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R!U!B'U R B DaR B R'B'D'aR'B'U F!D'F!R! 26,21 LV MG.1997 U'R B'R'U R'D!F U F U'F'D!U'R U 18,16 DU4V MG.1997 R!F'D'F!L F'R F R'aF'D F D'R! 18,15 DU5V MG.1997

The I's - semi-ornaments

U!R!D!U!R!D'a 12,7

The K's

There are 5 "K"-ornaments that can be obtained as a composition of "U"- ornaments and crosses-ornaments. Another 5 "K"-semi-ornaments can be obtained as a composition of "U"-semi-ornaments and crosses-semi- ornaments, and 2 "K"- simple patterns can be obtained from the one-edge-little-square simple patterns

D'aFsD!BsUsFsD!Bs BsU!FsR!U!R!U!R!

composed with a chessboard simple pattern.

Supplements of the 'K's

D'L'DsF R D'LsB L'R!B'UsR!DsB'R!D!R UsLsD'F' 32,28,22 [DLHH']

The L's - simple patterns

The first 14 types are members of the square group Q2. I tried to get algorithms minimal in q-moves and using a square group method too. Next 10 types are members of the group Q1. Q1 contains simple patterns which are not achieveable by square moves. For example

U'F R F!R'F'U!R'F'R!F R U' 16,13 U'F R F!R'F'U!F R F!R'F'U' 16,13 U'R F'R'F R F'R'F R F'R'F U 14

Q0 is the group generated by [F!,R!,B!,L!,U!,D!,(FUL,FDR)(FL,FR)]. Q0 consists of all positions whose colors of faces are two-color combinations of opposite faces of the Cube: (FB)(RL)(UD). Q2 is the group of positions generated by [F!,R!,B!,L!,U!,D!], It includes the three-cycle of edges R!U!L!B!R!U!L!F! , E2+2 L!F!L!D!F!L!F!D!. Q1 is a group of positions generated by [F!,R!,B!,L!,U!,D!,(FUL,BUR,FDR)] - Q2, Q3 = Q0 - Q1 - Q2 are odd positions. For example

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D'F!U L!U'F!D'R!U F!U'R!D 19,13 This pattern is not pretty one, of course

6 L's from Q2

There are four groups of relative 6L patterns (14=1+1+4+8). A short bridges of the type X!Y! exist between the members of the group.

The first of them has the most interesting symmetries.

UsBsR!FaD'aF'aD!Fa 16,14,12 L1 MG.1982 F!L!B!U!L!D!L!U!B!U!B!R! 24,12 p176 6 L FaL!F'aR!UsBsR!F'aUs 18,15,12 L2 L!D!B!D!B!L!B!L!B!D!R!U! 24,12 MG.1997 F'aD!BsLsD!Ra 12,10,8 L3 MG.1982 R!D!R!F!R!F!R!F!D!F!R!s 24,12 MG.1997 F!sL!D!L!B!L!F!L!F!D!B! 24,12 MG.1997 F'aUaL!DsBsR! 12,10,8 L4 MG.1982 R!D!R!F!R!F!R!F!D!F! 20,10 MG.1997 R!U!L!B!L!B!L!B!U!F! 20,10 MG.1997 FaD!BsLsU!R'a 12,10,8 L5 MG.1982 R!D!R!F!R!F!R!F!D!B! 20,10 MG.1997 R!U!L!B!L!B!L!B!U!B! 20,10 MG.1997 FaUaR!DsBsL! 12,10,8 L6 MG.1982 F!sL!D!L!B!L!F!L!F!D!F! 24,12 MG.1997 R!D!R!F!R!F!R!F!D!B!R!s 24,12 MG.1997

{L3} + R!s = {L4}, {L3} + F!s = {L6} {L4} + F!s = {L5}, {L4} + R!s = {L3} {L5} + R!s = {L6}, {L5} + F!s = {L4} {L6} + F!s = {L3}, {L6} + R!s = {L5}

FaD!L!Fa 8,6 L7 MG.1982 U!B!R!B!U!B!D!F! 16,8 MG.1997 D!F!L!B!U!B!U!B! 16,8 MG.1997 FaD!L!FaD!R! 12,8 L8 MG.1997 U!B!R!B!U!B!D!F!D!R! 20,10 MG.1997 B!L!U!L!BsD!Fa 14,9,8 L9 MG.1997

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D!F!R!D!F!D!F!D!F!D! 20,10 MG.1997 FaD!L!FaU!L! 12,8 L10 MG.1997 U!B!R!B!U!B!D!F!U!L! 20,10 MG.1997 D!F!R!D!F!D!F!D!F!R! 20,10 MG.1997 DsR!sDsF'aU!R!Fa 16,12,9 L11 MG.1997 U!B!D!sR!F!D!F!D!F! 20,10,9 MG.1997 DsL!UsFsR!F'aD! = L!UsB!RaF!R'aD'a 14,11,8 L12 MG.1997 U!B!D!sR!F!D!F!D!F!D!R! 24,12,9 MG.1997 U!B!R!B!U!B!D!F!D!R!U!s 24,12,9 MG.1997 LsU!sLsF'aU!R!Fa = BsD!RsF!U!F!R'aFa 16,12,9 L13 MG.1997 B!L!B!L!B!D!R!sF!R! 20,10,9 MG.1997 DsR!UsBsR!F'aU! = R!UsF!RaF!R'aDa 14,11,8 L14 MG.1997 = FaR!F'aR!UsF!Ua = FaR!F'aUaR!UsF! 14,11,10 MG.1997 D!F!R!D!F!D!F!D!F!R!U!s 24,12,9 MG.1997

{L7} +D²R²={L8}, {L7} +U²L²={L10} {L8} +R²D²={L7}, {L8} +L²U²={L9}, {L8} +U²D²={L12}, {L8} +F²B²={L8}^ {L9} +D²R²={L10},{L9} +U²L²={L8} {L10}+R²D²={L9}, {L10}+L²U²={L7}, {L10}+U²D²={L14}, {L10}+F²B²={L10}^ {L11}+D²R²={L12},{L11}+U²L²={L14} {L12}+R²D²={L11},{L12}+L²U²={L13},{L12}+U²D²={L8}, {L12}+F²B²={L12}^ {L13}+D²R²={L14},{L13}+U²L²={L12} {L14}+R²D²={L13},{L14}+L²U²={L11},{L14}+U²D²={L10}, {L14}+F²B²={L14}^

In this group ^ means F<->B symmetry.

A schema of short bridges:

Connections between L3, L4, L5 and L6 are of the type F!B!. Connections between L7,L8,L9,L10,L11,L12 and L13 are of the type F!R! and F!B!.

L7 R²s / \ L3 --- L4 = L8 - L9 -L10 = F²s | | F²s U²D² | | U²D² L6 --- L5 =L12 -L11 -L14 = R²s \ / L13

= ... F²B² bridge to its own mirroring

Now let us consider bridges of the type F!R!sB!. In this way, we get connections between L1, L2.

{L1} + U²R²sD² = {L2} http://ws2.binghamton.edu/fridrich/ptrns.html Page 8 of 23 Pretty patterns 11/25/11 10:58 AM

{L1} + U²R²sD² = {L2} {L1} + F²U²sB² = ru{L2} {L1} + R²F²sL² = dl{L2} {L1} + D²R²sU² = {L2} {L1} + B²U²sF² = ru{L2} {L1} + L²F²sR² = dl{L2} ru.ru=dl, ru.ru.ru=identity

L1 / | \ L2 L2 L2 | \ | \ | \ L2^ L2^ L2^ \ | / L1°

° ... center of the cube symmetry (FB)(RL)(UD)

We also get new connections between L3,L4,L5,L6.

{L3} + D²R²sU² = {L5}^ (F<->B mirroring) {L4} + D²R²sU² = {L6}^ (F<->B mirroring) {L5} + D²R²sU² = {L3}^ (F<->B mirroring) {L6} + D²R²sU² = {L4}^ (F<->B mirroring) {L3} + F²R²sB² = {L6}^ (R<->L mirroring) L3 L4 {L4} + F²R²sB² = {L5}^ (R<->L mirroring) | X | {L5} + F²R²sB² = {L4}^ (R<->L mirroring) L6 L5 {L6} + F²R²sB² = {L3}^ (R<->L mirroring)

{L7} + L²F²sR² = {L9}^ (F<->B mirroring) {L11}+ L²F²sR² = {L13}^ (F<->B mirroring) {L7} + F²U²sB² = rr {L9} {L9} + F²U²sB² = rr {L7} {L8} + F²U²sB² = rr {L8} {L10}+ F²U²sB² = rr {L10} {L11}+ F²U²sB² = rr {L13} {L13}+ F²U²sB² = rr {L11} {L12}+ F²U²sB² = rr {L12} {L14}+ F²U²sB² = rr {L14}

About inversions: {L1}' = {L1}^ any mirroring along axis FRU--BLD {L2}' = {L2}^ (B<->U)(F<->D) {L4}' = {L4}^ (F<->R)(B<->L) {L3}' is not 6 L's {L5}' = f²u{L5} {L6}' is not 6 L's {L7}' = {L7}^ (R<->U)(L<->D) {L8}' is not 6 L's {L9}' = {L9}^ (R<->D)(L<->U) {L10}' is not 6 L's {L11}' = {L13}^ (F<->B)(R->U->L->D) {L12}' is not 6 L's {L13}' = {L11}^ (F<->B)(R->D->L->U) {L14}' is not 6 L's

U R!U R!D!F!D R!F!U!F!U!F!D' 24,14 L'9 MG.1997 U R!U'B!D!L!U'F!U'F!D!B!D!F! 24,14 MG.1997 D!F!L!U!D B R B!R'B'D!R'B'R!B R D 24,17 L'10 MG.1997

{L'4} = {L'3} + RsF²LsD² f²d{L'4} = {L'4} + UsB²DsL² {L'5} = {L'3} + UsB²DsL² f²d{L'5} = {L'4} + BsU²FsR² {L'6} = {L'3} + BsU²FsR² f²d{L'6} = {L'5} + BsU²FsR² http://ws2.binghamton.edu/fridrich/ptrns.html Page 9 of 23 Pretty patterns 11/25/11 10:58 AM

{L'9} = {L'8} + LsD²RsF² f²d{L'9} = {L'10} + RU²sL²U²sR {L'10} = {L'8}+ FsL²BsD² f²d{L'10} = {L'10} + DsL²UsB² f²d{L'10} = {L'6}+ R²F²R²F²R²F²

About inversions:

{L'1}' = {L'1}^ any mirroring along axis FRU--BLD {L'2}' = {L'2}^ (F<->R)(B<->L) {L'3}' = {L'3}^ (F<->R)(B<->L) {L'4}' = {L'6}^ (F<->R)(B<->L) {L'5}' = {L'5}^ (F<->R)(B<->L) {L'6}' = {L'4}^ (F<->R)(B<->L) {L'7}' = {L'7}^ (F<->R)(B<->L) {L'8}' = {L'9}^ (U<->D) {L'9}' = {L'8}^ (U<->D) {L'10}'= {L'10}^ (F<->R)(B<->L) There are two cases for the 6L's with respect to the corners:

a) the two opposite corners are in place (L1,L2,L7,L9,L11,L13), (L'1,L'2,L'7). b) the two adjoining corners are in place (L3,L4,L5,L6,L8,L10,L12,L14), (L'3,L'4,L'5,L'6,L'8,L'9,L'10).

Only two types have the same orientation for all L's (L1,L6).

Regular ornaments

F D R D'L'D R'D'F'R'F L F'R 14 [V] L'D R'D'L D F L F'R F L'F'D' 14 [V] R'D!R B'U!B R'D!R B'U!B 16,12 [V] LsF L'FsU F'U'BsR 12,12,9 [H] R D'LsF'RsD LsF L' 12,12,9 [H] L'DsB D B'UsR D'Ls 12,12,9 [H] LsF!R!D'FsU'F!sDFsDR!F!Rs 24,18,13 [SS'] MG.1982 R!F LsF RsU'F'U'F'RsU LsU R! 20,18,14 [HH'] MG.1983 R'B L B D'B D FsL'B!F U F'U'F'R 18,17,16 [HH'] Nánásy.1983 U'L'F L'F'R'F LsF'L F R F'R U 16,16,15 [OO'] MG.1982 U'R F'R'F L F'RsF R'F'L F L'U 16,16,15 [OO'] Nánásy.1982 U L D R!U R U'R B'D B'D'B!D'L'U' 18,16 [VH] MG.1997 R U'R B'D B'D'B L B'U R'U R U!B!R!L'U 22,19 [VSS'] 2 peaks MG.+JF. R U'R B'D B'D'B L!U!F U F'U L'B R!L'U 22,19 [VSS'] 2 peaks MG.+JF. ? [VHH'] L!R'F D!L'F'RsFsL F U!L'B F! 20,16,14 [VOO'] crosses MG.1997 = C.ful.C', C = L!R'F D!L'F'Rs derived from p1

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R'F!sR!U!sL'BsL'F!sL!U!sR'Fs 28,18,12 [VOO'] MG.1983 R'F!sR!U!sL'BsR'F!sR!U!sL'Fs 28,18,12 [VOO'] B'L F'L F R'F'L F R B'R'B R F'LsB'R B!L F L' 24,23,22 [VOO'] MG.1982 F R!sFaR!sF L B!sRaB!sL U!s 28,18,13 [VOO'] p101 (Plummer's Cross) (K) ? [SS'H] U'F!U R U!R!U!R'F'R!F U'F!U 20,14 [HOO'] MG.1982 B'L'D L'D'L'F LsF'L F R F'L B 16,16,15 [HOO'] MG.1982 D L'B R D'R'D B'L B R'B'R D' 14 [HOO'] p87 Twisted Cube Edges

Regular ornaments - supplements

R'D!R B'U!B R'D!L DsBsR'F!L 20,16,14 [DV] MG.1982 U L F R'DsBsR'D R'D'F'R'F L F'R 18,18,16 [DV] MG.1982 BsU F'LsD F'U'BsR 12,12,9 [DH] B R B'R!U'F'L U R D L U'B'L U L! 18,16 [DSS'] p86 Twisted Rings LsF!R!D'FsU'F!sD LsDsB!F'U!R!Us 28,21,15 [DSS'] B'R'U'LsF R U!F R BsU'F'L' 16,15,13 [DHH'] snake R. Schoof = C.u.C'^, C = B'R'U'LsF R U U B!L D FsL'DsRsF'D!R' 16,14,11 [DHH'] p121 Improve RCC 3.9.31 U'R F'R'F L F'DsBsL U'R'D R D'F 18,18,16 [DOO'] Nánásy U'R'DsBsL D'R'U'R DsR'D R U R'U F 20,20,17 [DOO'] MG.1982 U L D R!U R U'L DsBsR'B L'B'L!B'D'F' 22,20,18 [DVH] MG.1997 L F L D'B D L!F!D'F'R U'R'F!D 18,15 [DVSS'] (cube in cube) = C.rf.C^, C = L F L D'B D L!F' U!F!R!U'L!D B R'B R'B R'D'L!U' 20,15 [DVSS'] p7a (K) F U!F'R'F!R!F'R U L D F'D'R!L'U'F U!F' 24,19 [DVSS'] MG.1982 F R B'L B R DsL'F'L UsB'R B R'U'R'U R'F' 22,22,20 [DVSS'] MG.1982 U F D'F!D F'R'F!R U'aB'U B!U'B L B!L'D 24,20 [DVSS'] U F'U!B U B'U!F U'LsB D'B!U B'U'B!D L' 24,20,19 [DVHH'] L!R'F D!L'F'DsBsD R F!D'L R! 20,16,14 [DVOO'] p1 6 X of order 3 = C.urb.C', C = L!R'F D!L'F'Ds R'F!sR!U!sR'DsBsU'R!sU!F!sD'Ls 30,20,13 [DVOO'] chessboard FsR!sU B!sU!R!sU L'U!sR!B!sL'Bs 32,20,13 [DVOO']p1a RsF'DsB'D!sF DsBaDsF D!sB'DsB'Ls 28,24,16 [DVOO'] MG.1982 ? [DSS'H] B'L'D L'U'BsLsU'R DsR'D R U R'D L 20,20,17 [DHOO'] MG.

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LsDsBsLs = DsBsLsDs = BsLsDsBs 8,8,4 [D] R'DsBsLsD = U'BsLsDsB = F'LsDsBsL 8,8,5 [D] L DsBsLsU'= D BsLsDsF'= B LsDsBsR' 8,8,5 [D]

Specials

F RsU!F!sU'aFsL'U R!sU!F!sD 26,18,13 chessboard of order 6 (K) p2 6 X of order 6

SEMI-REGULAR PATTERNS

Semi-regular ornaments

R'F'L BsD'F'D FsL'F!R 14,13,11 [S] (easy) R U!L'UsB U'B'DsL U'R' 14,13,11 [S] U F'U'F U RaU F U'F'R'aU' 14 [S] R F!B'D'S'fD!SfD'B F!R' 16,13,11 [S] R U F'U!sB U'B'U!sF R' 16,12,10 [S] p196 D R B'L!B R'B'L!B D' 12,10 [O] (easy) D'L B!L'F'L B!L'F D 12,10 [O] R'U F!D'F'D F!U'F R 12,10 [SO] (easy) R'F!L'D'L F'L'D L F'R 12,11 [SO] R'F'L F'D'F'D F L'F!R 12,11 [SO] R'F'L BsD'F'D B'L'B L F L'B'F!R 18,17,16 [SO'] MG.1982 D L U'F'L'B'U B L F R!B'R'B R'aD' 18,17 [SO'] MG.1997 U R'F!R F D'F!D F'U' 12,10 [HO] (easy) U F R'F R F D'F'D F!U' 12,11 [HO] U'L!D B'D'L D B D'L U 12,11 [HO] FsU'RsU'R!sD RsU Bs 16,14,9 [SH] MG.1982 B!R U!B!L'B'L B'U R'U B! 16,12 [VS] JF.1982 B!R U'R B'R'B!U'B'U R'U B! 16,13 [VS] B!F'U F U'R U'B U'B'U R'U B! 16,14 [VS] B!F'U F U'R U'R'U R'F R F'B! 16,14 [VS] B'L F'L!FsR'B R F'LsB'R B!L F L' 20,18,16 [VO] MG.1982 F D R D'L'B'U'B D B'U B R'D'F'R'F L F'R 20 [VO] LsU FsD F!sU'FsU'F L'FsU F'U'BsR 24,22,16 [SHH'] MG.1982

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R U F'U!B U'FsD F'U!F D'R' 16,14,13 [SOO'] Nánásy.1982 R'F'L F'D'F'D L'B L F L'B'F!R 16,15 [SOO'] MG.1982 R'F!L'D'L F!B D B'D'FsL B L!D L F'R 22,19,18 [SS'O] MG. R'D B D B R'U!R B'D'R!B'D'R 16,14 [SOH] MG.1982 R'D B R'U'B L U!L'B'U R'B'D'R 16,15 [SOH] U'FsUsR F!R'DsR'B R!F'R U 18,16,13 [SOH] R'F'L FaD'F D FsL'U F!U'R 16,15,14 [SO'H] Nánásy.1982 R'F'L BsD'F'D B'L'F U F U'F R 16,16,15 [SO'H] Nánásy.1982 B D L'D!R D'FsUsR U'L!U L'D' 18,16,14 [VSH] MG.1982 R'D'R!B R'F'R B'R!F R'D R F R F' 18,16 [VSO] MG.1982 ? [VSO'] BsR'D B!D'R F!sU L'B!L U'Bs 20,16,13 [OHH'] B!D L'F'D!F L D'L'D F L!F'D'L B! 20,16 [OHH'] B!D L'F'D!L'F U L!U'F'L F L D'B! 20,16 [OHH'] MG.1983 B D R'B'D L F'D'L B!L'D F L D'B R D'B' 20,19 [OHH'] D'L F R U!L U!R!U!L'U R!U R'F'L'D 22,17 [VHO] MG.1982

Semi-regular ornaments - supplements

R F!B'D'BsRaDsF'R!L U' 16,14,12 [DS] MG.1982 B'D L!D'R'D L!U'BsLsD L 16,14,12 [DO] MG.1982 D'L B!L'F'L B!R'DsBsL B 16,14,12 [DO] MG.1982 R'U FaLsDsF'R'B R!F'R U 16,15,13 [DSO] MG.1982 R'F'L F'D'F'D F R'DsBsRaU 16,16,14 [DSO] MG.1982 R'F'L BsD'F'D B'L'B L F R'DsBsR U 20,20,17 [DSO'] MG.1982 U R'F!L DsBsL B'R!B R'F' 16,14,12 [DHO] MG.1982 U R'F!R B LsDsF'R!B R'F' 16,14,12 [DHO] MG.1982 FsU'RsU'R!sD!U'BsLsF Ls 20,17,11 [DSH] MG.1982 BsLsD U!RsU'R!sD RsU Bs 20,17,11 [DSH] MG.1982 B!R U!B!L'B'L F'LsDsB U'F L! 20,16,14 [DVS] MG.1982 B!R U!B!R'DsBsR'D L'F U'F L! 20,16,14 [DVS] MG.1982 B'L F'R'aDsBsU'L U R'DsL'U L!D R D' 22,21,18 [DVO] MG.1982 F!L'F R UaF'aRaD'L D'L!R'U!F!L 22,18 p20 Mark's Pattern 1 (K) LsU FsD F!sU'FsU'F R'DsB R'F'LsU 24,22,16 [DSHH'] MG.1982 R'F'L F'D'F'D L'B L F R'DsBsR U 18,18,16 [DSOO'] MG.1982 ? [DSS'O]

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R'BsL F!R'DsR'B R!F'R U 16,14,12 [DSOH] Nánásy.1982 U'B L B L U'F!U L'B'U!L'F'LsDsB 20,18,16 [DSOH] MG.1982 R'F'L B!LsDsF'R B RsD'F R!F'U 20,18,15 [DSO'H] Nánásy.1982 D L B'L B L U'F L'F'L'U L'D' 14 [DVSH] MG.1982 D L U'L!U L'FsL F'L!B L'D' 16,14,13 [DVSH] MG.1982 R'D'R!B R'F'R B'R!F R'D L DsBsL U R 22,20,18 [DVSO] MG.1982 R'D B D L'FsU B U'BsLsU!R B'D'R!B'D'R 24,22,19 [DVSO] Nánásy.1982 ? [DVSO'] B'L!F L'D'B'U RsB'D F D'B L B L'B D 20,19,18 [DOHH'] MG.1982 ? [DVHO]

Ornaments composed from U2, U3 or U4

D F'UsL'DsF UsL!F'L DsF'LsD BsR'D B'R' 26,25,19 [DU3U4] U L F'L DsF'LsD BsR'D LsU'L'U'F'aU'BsL F!U 28,27,22 [DU2U3]=[LU3]

Semiregular semi-ornaments

Even positions

RsD!F R!DsB!R BsD!L!U F!sU' 24,17,13 [WOS(1324)S'(14)(23)] (K) p138 Mark's Pattern 5 D'F'D F DsB L'D L UsB'D'B R'F'D'F'D 20,20,18 [WORKS(14)] MG.1982 B R D!L'D R!D'L D!R'B R!B! 18,13 [WRR'] U'F!U F!R'B'U'F!sD B L'F!U'R!D 22,16,15 [WRR'S(1423)S'(1423)] p158 Mark's Pattern 14 U'B!RaU!R'B'U L'B UsR'B U'R B'DsB!R!U 26,22,20 [WS(14)S'(23)] Walker B'U'FaR U'R!U R'F'aU B 14,13 [OKK'S(14)] B'D'RaB L'F!L B'R'aD B 14,13 [OKK'S(14)] L'DsF D'R!D F'UsL 12,11,9 [ORKS(14)] D R!U'aF'aR!FaR!U 14,11 [RK] D R!sU!B D!R!sU!F D 18,11,9 [RK] p181 Mark's Pattern 18 R'(D B D'B')^3 R 14 [RR'] FaR!sF'aU'F!UaF!U D! 18,13,12 [RR'S(14)(23)] D'F!D'aR!sB!R!sD!U 18,11,9 [RR'S(14)(23)] D FaR!F'aD!R'aB!RaD 16,13 [RR'S(14)(23)S'(14)(23)] http://ws2.binghamton.edu/fridrich/ptrns.html Page 14 of 23 Pretty patterns 11/25/11 10:58 AM

4 T's type 1 Nánásy.1982 D'B!RaF!UaB!U'aB!R'aD 18,14 [RR'KK'S(14)S'(23)] (F!U F!sU B!)!Us 22,14,10 [RR'KK'S(1342)S'(24)] p163 4 ARM Full U'L'B'U'R!sD F L'R!D!F!U!sF!U' 24,16,14 [RR'KK'OS(12)] p147 ML's 6 ARM Pattern (K) U RaF!R'aU' 8,7 [RKS(14)(23)] D FaR!F'aD' 8,7 [RKS(14)(23)] U R'aB!RaU' 8,7 [RKS(14)S'(23)] D F'aL!FaD' 8,7 [RKS(14)S'(23)] U'R F!sL!F!sR'U 14,9,7 [RKS(14)S'(14)] D L!D'aBaL!B'aL!U = D'RaB!R'aB!UaB!U' 14,11 [RKS(23)S'(23)] U'R!UaR!D' 8,6,6 [KK'S(14)(23)] D'F!DaR!sB!R!sU' 16,10,8 [KK'S(14)S'(23)] U'FaR!F'aU!R'aF!RaU' 16,13 [KK'S(14)(23)S'(14)(23)] 4 T's type 2 Nánásy.1982 RsDsR UsF!LsD = SrSfD SbD!SlD 12,11,7 [S(13)S'(24)] (U L!B!R!D)!L!R!B!F! 24,14,12 [S(13)(24)S'(12)(34)] p165 2 Swap, 4 H full D'R!F'aD!FaR!F!D 14,10 [S(14)(23)] D F!R!F!D!R!F!R!D 16,9 [S(14)(23)] D'R!FaU!F'aR!F!D 14,10 [S(14)S'(23)] Sf!Sr!D Sf!D!Sr!D 20,11,7 [S(13)(24)S'(13)(24)] Sr!Sf!D Sf!D!Sr!D 20,11,7 [S(13)(24)S'(13)(24)] U'R!UaR!sD'aL!U 14,10,9 [S(14)(23)S'(14)(23)] (D R!U )^4 16,12 p162 2 X, 4 H full (F!sD F!sU)! 20,12,8 [S(134)S'(134)] p170 2 edge swap + 2 H R'B R!L'DsFsU RsB'D'L'D Rs 18,14,11 [DS(14)] RaF!sR'aDs 10,8,6 [DWO] (D R!FaU)^3 U!sR!U!sL! 30,21,19 [DWORK] p178 2L, 2 Fours, 2 arm D'L!FaD!F'aL!D!sF!U!D' 20,13,12 [DWOS(14)(23)] U!F!D!U!F!Da = U F!U!D!F!D!U 12,7 [DWOS(12)S'(12)] U F!D!sF!D!U 12,7,6 [DWOS(14)S'(14)]

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F!U!RsD!sLsU!B!Us 18,12,8 [DWOS(12)(34)]S'(12)(34)] U!F!R!sD!R!sD!F!Da 20,11,9 [DWOS(12)(34)]S'(12)(34)] B'U'R F'U RsB'U R'B U'RsFaUs 18,18,15 [DO] R D'B L'D F!sU'R B'U L D!sR!Us 22,17,14 [DO] U R'B'U'F!sD B L'D F!R!F!U'F!D' 22,16,15 [DORR'] (K) p93 Mark's Pattern 2 R F U FsR'F'R F'R DsB'L'R!D'F' 18,17,15 [DOS'(143)] MG.1982 D R!UsB!R!B RsD'R!D LsB'U' 20,16,13 [DOS(14)S'(14)] MG.1981 D R!UsB!U'R!B RsD'R!D LsB' 20,16,13 [DOS(14)S'(14)] p81 Snake type 1 D R!UsB'LsB'R'B RsD'R D B'U' 18,17,14 [DOS(14)S'(14)] MG.1981 D R!UsB RsB R B'LsU R'U'B U' 18,17,14 [DOS(14)S'(14)] MG.1981 F!sLsB'DsL!F!L'FsU B!D' 20,15,11 [DS(13)(24)S'(13)] (K) p146 Mark's Pattern 9

Odd positions

R D'L!B L B'L U!R'F R U!D R' 17,14 [OK] R!F!D F D'F L!B'U B R!s 17,12,11 [KS(14)] L!F L'D'L F'L'D F'D'F'D F L' 15,14 [KS(14)] L F'D'aF D F'U F!D'F D F'L' 15,14 [KS(14)] F!D F!D L!D F!D'F!D'L!D!F!D'F!D' 25,16 [KS(14)] R!U'F'U F'L!B D'B'L!F!R! 17,12 [KS(13)] F'D'F L!B'U B L!F!D F! 15,11 [KS(12)] RaDaR'D'R U'R!D R'D'Rs 15,14,13 [KS(23)] D!B!R!D'R'D R'B!L U'L'D! 17,12 [RS(13)] U!F!L!D'L'D L'F!R U'R'U! 17,12 [RS(13)] U R'F R'B'R'B R'F'U'R U R'U! 15,14 [RS(13)] U R'B U'B'U'B U R U R'B'R U! 15,14 [RS(13)] U!L!U B!U'B!D L!D'B!D B!U! 21,17 [RS(1324)] R D'F!D F'R F UsF DsF'R!D F D'R D F'D' 23,21,19 [ORKK'S(14)(23)] MG.1983 D'RaB!L'U'aL!B L B'L U!R'F R DsR'D 23,20,19 [ORR'K'S(23)S'(14)] MG.1983 F!D!sF R'D R!D'R F'U R!U!R!D!F! 25,16,15 [ORR'K'S(23)S'(14)] (K) p119 Mark's Pattern 3 R U'F R'B'D!R'U'BsLsD B L U'R'U'F U'Ls 23,22,19 [DK]

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Cyclic patterns

R!sU F!sU!R!sD'F!sD! 22,12,8 crosses R!sD!F!sD R!sU!F!sU' 22,12,8 crosses U R!sU'R!sU!R!sD'R!sU' 22,13,9 crosses + 2 X's R'B'RsF'R'SdR F LsB R 14,14,11 4 U's MG.1981 F R'B'D!L'UsB D!R F L' 14,12,11 p112 4 U order 4 R'B'RsF'R'U R F LsB R 13,13,11 4 T's MG.1981 R!sF!sD'F!sR!s 17,9,5 4 Y's D R!sDsR!sU'a 13,8,5 4 Y's + 2 X's FsL F!L!UsB U!RsB!D L!U' 20,15,12 p114 3 ARM Order 4 U'B'R'U F'UsL U'F R'U R!sD'L!D 20,17,15 4 X's & 2 K's (K) p122 Mark's Pattern 4 D'R F R'F!B L D'F U L'B R!sF!R 20,16,15 4 O's & 2 K's p144 Mark's Pattern 7

Three-color regular patterns

R!sUsRsBsUs 12,10,5 blossomes B R'D!R D'B!D R'D!F!R U'B!U R'F!R F (F) 24,18 6 Stripe type 1 U F B R DsRsD'R'aB R!D!R!B!R 21,17,15 p120 6 Stripe (1) L U R'B!R B'U!B R'B!D!R F'U!F R'D!R U'R 26,20 6 Stripe type 2 RaD R'aD F'aU'FaRaU'R'aD'F'aD Fa 22 diagonals (4 Peak flip) 21,16,14 U'F!L!B!D!F!R'UaF'LsBsR U' Tetra Peak exch (K) p69 D'R'BULsB'UB'U'BL'BLB'UB'U'BRsDB'DsF'UsLB 30,30,26 giant meson 1 L D R!D'L!U B'D'B D'R'D R DsL D'B!D 22,19,18 giant meson 2 R!L'DBR'D'BLBL'D'B'D!LDL'U'FD'F'RFL'FLF!R'D!U some about g. meson 1 DFRsU'B'D'R'aD'LsBDFD!F!DF'R'B'L'DLBF!RD'R!D some about g. meson 2 4 Peak flip + 4 twists RaDaR'aF'aDaFa (Pinwheels) U'L'B L'B'L'D!U'FsRsF'U F D F'U B 20,19,17 [D'HOO'] (see [HOO']) LsD L'DsF D'B'RsD!U' 14,13,10 [D'H] F R!sU!sF!B'R!U F!sR!sU!D'L! 28,16,12 p82 8 Crn Twist 1 U R U'R D!B!U'L'D R FsUaL!U!sB! 24,18,16 [DV]+chessboard s.p. (K)

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p125 ML's Multicolor 1 B!D'F!U'L!D!F!U'F R B U!F!R F!D FsU'F' 28,20,19 Triple Threes (K) U B!D'L!F!R!U'B U'L D F!L U'B'F'R!U'F! 26,19,19 Orthogonal bars (K) UsRsFsU L UsB'DaB'D!sF U!R U!sL'D!U 30,24,18 (similar to [D'HH']

Three-color regular ornaments

B'L'D L'D'L'F R FsUsF'D'F RsU'F B!D 22,21,18 [D'OHH'] (see [OHH']) B!R!D!R!B'D!B'L!F U!L'D L D!R'F L!F!D B' 30,20,20 Treep's Skein (K) U!B'D!B'U!B!U!B'D!B R!L B R D'R F!D U!R B 30,21,21 Z-snake type 1 (K)

More-colored regular patterns

F!D'F!L!U'F!R!F!D F'U!LsFsU Fs 25,18,15 Perry's Pinwheel (K) D!B!R!F!R!D!U L!F!D RaF'L F'aD'B!L!U' 30,20,20 Six L's - 6 colors (K) U!L!U L!D'L!U B!U'F'aRsF'D!L!U'B'D B 27,20,19 Two diamonds (K)

More-colored nonregular patterns

B R'D!R D'B!D R'D!F!R U'B!U R'F!R B' 24,18 4 Stripe D RaD!F LsBsR'D'aB U!RsB! 20,17 p92 4 Stripe (Full) R'B'L!BD'R!D B'L!U!B U'R!U B'U!B R 24,18 4 Stripe + 2 X's MG.1982 R!F!U'L!U!F!U'F!L!F!R'F R!B L F'R!B'R D!R! 33,21,21 Four T's (K) R'U'B'R!B R B U'FsU F'L'F'L!F U L 20,18,17 2 Peak twist MG.1983 U R'F'R!F D R U'R D'R!L'U'RaFaU'F'a 22,20 3 Peak twist MG.1982

Flips and twists

(U R'F R )^5 20 p171 6 Edge Flip (F R'D'R)^3 (L'B D B')^3 24 6 flip (RaUaFa)! 12 8 edge flip R U'R!U'R U'aR U'R'D!R'U'R U'a 18,16 p183 6 Twist U F!L F!sU!sR'L!B!U!D'F!sR!s 28,16,12 p155 8 Crn Twist 2 (K) (L R!F!B')^4 24,16 p156 8 Crn Twist 3 RaD!B'L!F!R!DsR'D!F'aD'F!D'R!U'F!D' 28,20,19 p3 12 flip (K) D F!U'B!R!B!R!L B'D'F D!F B!U F'RaU!F' 28,20 p6 12 flip, 8 twist (K) FsR UsB!R!B'LsF!sU'F!D'F!sD'R!D' 28,20,15 p139 6 X + 12 flip (K)

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B'L D L F'U'aL B'D'F'RsF'UaL!UaB! 22,20,19 p139a (K) F R'DsR'B D'F!R!L F R'B'L D!B!U'R!sD'R!s 30,22,19 p141 Superfliptwst + 6X (K) F R L!U'R!L'U'D!R!F D B D F!U'R'D'F!D!L! 28,20 p141a (K)

Patterns possible by disassembly

U L U L'U'F L'U'F'U'F'U L U L!U'F'L' + FLU+ 19,18 2 Peak exch MG.1983 R'D!B!L B L'B D'R D' + BDR- 12,10 1 Peak twist MG.1982 R'D R'B U'B U B!R!D' + BDR- 12,10 MG.1982 R'D R'B R B!D B D'R D' + BDR- 12,11 JF.1983 R'D R'B R B'R D'R F'R'F + BDR- 12 MG.1982 F D'F'D R'D B'D B D'R D' + BDR- 12 MG.1982

Ornaments

U'R D BsR'U'L Ds + (UR,LF+) 10,10,8 3 U's + 3 O's MG.1982 L'UsL'U R FsD'R'U B + (UF,BL+) 12,12,10 3 U's + 3 O's MG.1982 D'R'aD RsF'aUsR FaL + (UR,BD+) 14,14,12 3 U's + 3 O's MG.1982 F U RsB'D'F'RsF'LsD F + (RD,RF) 14,14,11 6 U's type 1 MG.1982 U'F U RsB'D'F'RsF'LsD F R + (UR,FR) 16,16,13 6 U's t2 MG.1982 U'F U RsB'D'F'RsF'LsD F R + (UR+) 16,16,13 3 U's + ring MG.1982

Regular semi-ornaments

L D R!D'L!B'D'B UsR'D R DsL D'B!D +(LDF,LFU) 22,19,17 cube within a cube

Semiregular semi-ornaments

R F U F!U'R'F' + (FD,RD+) 8,7 [VRKS(14)] U'B!L'F D'F'L B!R U R + FU+ 13,11 [VRS(14)] F!R'D R D'R F!D R'D'R' + (FD,RF) 13,11 [RS(14)(23)] F!R'D R D'R F!D R'D'R' + (FD,FR+) 13,11 [VRS(14)(23)] R F U F!U'R'B'D'F'D BsR'F R + FD+ 16,15,14 [VRKS(14)(23)] D!F UsL'D!L!FsU!D FsL' + (BL,DL) 18,14,11 [D] R'F L DsBsD LsB'U'R'U Ls + (DF,UL) 16,16,12 [D]

Cyclic patterns

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RF'DR'U'B'UB'R'B'RBR'D'R'FR'Us + (LUB,LBD) 19 O's MG.1982

B'D R D'RsF R'F'R'F'UsL B R + (BU,LU) 16,16,14 O's MG.1982 F L B UsR'U'B'U FsR'B R F'L' + (BU,LU) 16,16,14 O's MG.1982 R'B D F'RsU F'D'F'D'LsB U F + (LU,LB) 16,16,14 O's MG.1982

Simple patterns

It is quite a qood excercise to try to get back the Cube from these positions. The number at each sequence means a number of 'long moves' like Fs,R!,Da. Three characters in the column 'position' describe what you can see on faces F,R,U. Only position with O,H,I,X,+,_ on all faces, where O,I,+ occur in pairs, (in parentheses in the column 'position') are possible.

slice moves position R!D'sR!sD'sL! 6 (H,H,H) FaR!F!sD!sL!F'a 8 (H',H,H) R!sF!sD!s 6 (X,X,X) RsFsRsFsRsFs 6 (X,X,X) RsF!sD!sRs 6 (X,X,X) R!sF!R!sB! 6 (_,_,H) D!RsF!sR'sU! 6 (_,_,H) DaF!sD'sF!sD! 7 (_,_,X) R!sF!R!F!sR!F! 8 (_,_,X) R!sD'sF!sD's 6 (O,O,_) RsD'sF!sD'sR's 6 (O,O,_) D!FsD!sFsD! 6 (O,O,H) D!R!sF!sD! 6 (O,O,X) R!F!DsF!sDsB!R! 8 (H,H,_) RaD!sR'aF'aD!sFa 8 (H,H,_) (F!B!R!L!U!)! 10 (H,H,_) p160a D!sRsF!sR's 6 (H,_,H) F'sR'sFsR'sF'sRs 6 (H,_,H) FsD!F!D!sF!D!Fs 8 (_,H,H) R!B!R!sB!L! 6 (I,I,_) R!s 2 (I,_,I) R!F!R!sF!R! 6 (_,I,I) U!RaD!sRaD! 6 (_,I,I)

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R!F!sR! 4 (I,I,H) R!FsD!sF'sR! 6 (I,H,I) DsFaD!sFaDs 6 (I,H,I) RaD!sRa 4 (H,I,I) R!B!DsF!sD'sB!R! 8 (H',I,I) F!D!R!sF!sU!B! 8 (H',I,I) R!sF!s 4 (I,I,X) RaD!sRsD!sR! 7 (I,X,I) R!FaR!sFaR! 6 (X,I,I) R!sDsR!sDs 6 (H,H,X) RsFsR!sFsRs 6 (H,H,X) R!F!sD!sL! 6 (H,X,H) R!F!sR!F!D!sB! 8 (X,H,H) DsRaU!R'aD'sF'aU!Fa 8 (+,+,_) FaUaR!sUaFaU!s 8 (+,+,_) p16a R!F!R!DsF!R!F!Us 8 (+,+,_) p16c R!D!sF!D!sF!R! 8 (+,+,H) D!sR!D'sR!sD'sL! 8 (+,+,H) R!sD'sR!sDs 6 (+,+,X) RsF'sR'sFsRsF's 6 (+,+,X) R!F!R!sF!L!D!s 8 (X,X,_) D!sR!F!sR! 6 (X,X,H) RaD!R'aFaD!F'a 6 (U,U,_) RaD!R'aF'aU!Fa 6 (U,U~,_) D'aB!DaL!D!R! 6 (U,U',_) R'aD!RaF!D'aR!DaF! 8 (Z,Z,_) R'aD!RaF!DaR!DaB! 8 (Z,Z',_) R'aF!D'aL!DaF!RaD! 8 (Z',Z',_) D'R'aD RaU!R'aD'RaD' 9 (Z',Z',_) F!sDaF!sD'a 6 (_,H,X) R!sF'aD!sF'a 6 (_,H,X) L!F!R!sF!D!sL! 8 (H,X,_) DaF!sD'a 4 (_,O,I) D!R!sU! 4 (O,_,I) U!R'sD!sRsD! 6 (_,O,+)

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F!R!sF!R!s 6 (_,I,+) F!R!DsR!sD'sR!B! 8 (I,+,_) DsF!sDs 4 (H,O,I) D!R!sF!sD!F!s 8 (O,H,I) D!R!sU!R!F!sL! 8 (O,I,H) F!RaD!sRaF! 6 (H,I,O) RsF!D!sB!R's 6 (H,O,+) DaF!D!sR!sB!Da 8 (O,+,H) R'sD!sRs 8 (H,I,+) R!F!sD!sL!D!s 8 (+,I,H) RsU!F!D!sF!U!R's 8 (+,I,H) R!D'sR!sDsR! 6 (I,+,H) R!F'sR!sFsR!D!s 8 (+,H,I) R!sF!D!sB! 6 (X,O,I) F!U!F!sU!R!sB! 8 (X,I,O) R!F!sR!F!D!sF! 8 (X,O,+) FaR!sFaR!s 6 (X,I,+) DsFaD!sFaD's 6 (X,+,I) F!L F!U!sB!R B! 14,8 4 T's F!L B!U!sF!R B! 14,8 4 T's

One little square of opposite color

RsU!RaFaU!F'aR! 14,11,10 [E4] L!D S!rD!S!rD L! 16,9,7 [E4] D!F!U!L!F!L!U!F!L!D! 20,10 [E4] RaD'aB!D'aR!D!Ls 14,11,10 [E6] U R!F!sL!D'BsL!BsD! 18,12,9 [E6] F!U!L!D!F!L!F!D!F!U!F!L! 24,12 [E6] B!U!R!B!R!B!L!B!U!B!R!s 24,12 [E6] B!U!B!L!B!R!B!R!U!B!R!s 24,12 [E6] R D'L B L!D!R F R'D!L B'D R' 17,14 [C2E2] D'L!F!U F!U'F!L!U'L!D F!D'L!D 23,15 [C2E2] Domino alg. U R'F'R!F R U!R'F'R!F R U 16,13 [C3]

Two corner little squares of opposite color

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RaD!B!D!B!D!LsB!L! 18,11,10 [C3+3] FaU!FsR!U!R!U!R!F! 18,11,10 [C3+3] F!D!R!D!F!D!F!R!F!D! 20,10 [C3+3] D'aFsD!BsUsFsD!Bs 16,14,9 [C2+4] R!F!D!R!F!D!R!F!D! 16,14,9 [C2+4] BsU!FsR!U!R!U!R! 16,10,8 [C6] U!R!U!F!U!F!R!F!U!F! 20,10 [C6]

Odd simple patterns

F'D!R!F R F'R U!L'B L D!sF D! 21,15,14 R B'D RaD'L'D R'D!L D'L'D B R'D! 19,17 F!R!U!F!R!U'R!D!B!U'L!U!L!B!U 27,15 p123 Cube in a cube 2

Algorithms signed (p..) where .. replaces a number were taken from Michael Reid's PATTERNS.TXT - August 24, 1996.

Algorithms with (K) as the author were probably produced by a computer program based on Kociemba's 2- step algorithm. The authors of those algorithms are not known to us. Please, if you know who obtained those algorithms for the first time, let us know. Thank you.

Algorithms signed (MG.) are Mirek Goljan's results, JF. denotes Jessica Fridrich.

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